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 Jan-Bernhard Kordaß committed Oct 25, 2016 1 2 3  \tableofcontents  Jan-Bernhard Kordaß committed Nov 22, 2016 4 \chapter{Torsion Invariants [Roman Sauer]}  Jan-Bernhard Kordaß committed Oct 25, 2016 5 6 7 8 9 10 11 12 13 14 15  Torsion invariants fall into a class of so-called secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of primary invariants'' (e.g. Betti numbers) vanish. Often they reveal more subtle geometric information. The following will contain a discussion of Whitehead and Reidemeister torsion. Informally, corresponding primary invariants are Lefschetz numbers (Whitehead torsion) and the Euler characteristic (Reidemeister torsion). \section{Review of Euler characteristic and Lefschetz numbers.} \subsection{CW Complexes} \begin{dfn*}  Jan-Bernhard Kordaß committed Nov 23, 2016 16  A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space homeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$.  Jan-Bernhard Kordaß committed Oct 25, 2016 17 18 19 20 \end{dfn*} \begin{expl*} \begin{enumerate}  Jan-Bernhard Kordaß committed Nov 23, 2016 21  \item Simplicial complexes, e.g. triangles, pyramids, etc.  Jan-Bernhard Kordaß committed Oct 25, 2016 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75  \item But CW-complexes are more general, the following graph is CW for example: \begin{center} \begin{tikzpicture} \draw (0,0) to[bend left] (2,0); \draw (0,0) to[bend right] (2,0); \draw (2,0) to (3,0); \end{tikzpicture} \end{center} One can even attach a disc along its boundary to a single 1-cell. \end{enumerate} \end{expl*} \subsection{Euler characteristic} \begin{dfn*} The Euler class $\chi(X)$ of a finite CW-complex $X$ is defined as $\chi(X) = \sum_{i \geq 0}(-1)^i \#(i\text{-cells of } X) \in \Z$. \end{dfn*} \begin{thm*}[Euler-Poincaré] \begin{align*} \chi(X) = \sum_{i \geq 0} (-1)^i b_i(X), \end{align*} where $b_i(X) = \rk_{\Z} H_i(X;\Z)$. \end{thm*} In particular, $\chi$ is a homotopy invariant. \begin{proof}[Proof''] $H_i(X;\Z) = H_i(C_{*}^{CW}(X))$, where $C_{*}^{CW}(X)$ is the cellular chain complex \begin{align*} \cdot \to C_{i+1}^{CW}(X)\xrightarrow{\partial} \underbrace{C_{i}^{CW}(X)}_{\cong \Z^{\# i\text{-cells}}} \xrightarrow{\partial} C_{i-1}^{CW}(X) \to \cdots \end{align*} Thus $\chi(C_{*}) := \sum_{i \geq 0} (-1)^i\rk_{\Z}(C_{i})$ and $\chi(C^{CW}(X)) = \chi(X)$. This boils down to \begin{align*} \chi(C_{*}) = \sum_{i \geq 0} \rk_{\Z}H_i(C_{*}) ( = \chi(H_{*}(C_{*}))]. \end{align*} This is just additivity of the rank! Consider \begin{align*} C_1 \xrightarrow{\partial} C_0 \end{align*} and note that we have the exact sequences $0 \to \Im \partial \to C_0 \to \underbrace{H_0}_{= C_0/\Im \partial} \to 0$ and $0 \to \underbrace{H_1}_{= \Ker \partial} \to C_1 \xrightarrow{\partial} \Im \partial \to 0$. Thus $\chi(C_{*)} = \rk_{\Z} C_0 - \rk_{\Z} C_1 = \rk_{\Z} \Im \partial + \rk_{\Z} H_0 - \rk_{\Z}H_1 - \rk_{\Z} \Im \partial = \rk H_0 - \rk H_1$, which completes the proof''. \end{proof} \subsection{Review of cellular homology} Let $X$ be a CW-complex with cellular decomposition $E$. Then we can consider the \CmMark{n-skeleton} \begin{align*} X^n := \sum_{e \in E, \dim e \leq n} e, \end{align*}  Jan-Bernhard Kordaß committed Nov 23, 2016 76 which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a push-out diagram  Jan-Bernhard Kordaß committed Oct 25, 2016 77 78 79 80 81 82 \begin{equation*} \begin{tikzcd} \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\ \coprod D^n \ar{r} & X^n \end{tikzcd} \end{equation*}  Jan-Bernhard Kordaß committed Nov 23, 2016 83 One could take this as an alternative definition of a CW-complex by a filtration with the push-out property.  Jan-Bernhard Kordaß committed Oct 25, 2016 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 The cells can be recovered as connected components of $X^n\setminus X^{n-1}$. We have \begin{align*} C_i^{CW}(X) = H_i(X^i, X^{i+1}) \xleftarrow{\cong} H_i(\coprod D^i, \coprod S^{i-1}) \cong \bigoplus H_i(D^i, S^{i-1}) \cong \bigoplus \Z^{\# i\text{-cells}}, \end{align*} where the first isomorphism $\leftarrow$ is given by excision. The boundary maps $C_i^{CW}(X) \xrightarrow{\partial} C_{i-1}^{CW}(X)$ come from \begin{align*} H_i(X^i,X^{i-1}) \to H_{i-1}(X^{i-1}) \to H_{i-1}(X^{i-1},X^{i-2}). \end{align*} Under this isomorphism, the matrix entry belonging to $(e,f)$ where $e$ is an $n$-cell, $f$ an $(n-1)$-cell is the \CmMark{degree} of the map. \begin{align*} S^{i-1} \xrightarrow{\phi_e|_{S^{n-1}}} X^{i-1} \xrightarrow{\operatorname{proj}} X^{i-1}/(X^{i-1}\setminus f) \xleftarrow{\phi_f, \cong} D^{i-1}/S^{i-2} \cong S^{i-1}. \end{align*} \begin{expl*} Consider the torus as an identification square. We convince ourselves that the cellular chain complex is given as $\Z \to \Z \oplus \Z \to \Z$, where $1 \mapsto (0, 0)$, since it is described by a map $S^1 \to S^1$ traversing the 2-cell according to orientation has degree $0$. \end{expl*} \subsection{Lefschetz number} Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, if $f(X^i) \subset Y^i$ for all $i$. \begin{thm*}[Cellular approximation] Any map between CW-complexes is homotopic to a cellular map. \end{thm*} \begin{dfn*}  Jan-Bernhard Kordaß committed Nov 23, 2016 115  The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defined as  Jan-Bernhard Kordaß committed Oct 25, 2016 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144  \begin{align*} \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z. \end{align*} \end{dfn*} \begin{rem*} $\Lambda(\id_X) = \chi(X)$. \end{rem*} The following theorem yields a description of Lefschetz numbers by homology. \begin{thm*} $\Lambda(f) = \sum_{i \geq 0}(-1)^i \tr H_i(f)$. \end{thm*} Thus, this number only depends on the homotopy class of $f$. \begin{proof} Similar to the proof of Euler-Poincaré using the additivity of the trace, i.e. in the situation \begin{tikzcd}[row sep=small] 0 \ar{r} & A \ar{r} \ar{d}{a} & B \ar{r} \ar{d}{b} & C \ar{r} \ar{d}{c} & 0\\ 0 \ar{r} & A \ar{r} & B \ar{r} & C \ar{r} & 0 \end{tikzcd} we have $\tr(b) = \tr(a) + \tr(c)$. \end{proof} \begin{thm*} If $f$ has no fixed point, then $\Lambda(f) = 0$. \end{thm*}  Jan-Bernhard Kordaß committed Oct 25, 2016 145   Jan-Bernhard Kordaß committed Oct 25, 2016 146 147 148 \begin{rem*} The converse is not true (think of counterexamples, e.g. $S^1 \wedge S^1$), although there is one in the case of simply-connected closed manifolds. \end{rem*}  Jan-Bernhard Kordaß committed Oct 25, 2016 149   Jan-Bernhard Kordaß committed Oct 25, 2016 150 151 152 153 154 155 156 157 \begin{proof} Let $X$ be metrizable and let $d$ be a metric. If $X$ is compact, there exists an $\varepsilon > 0$ with $d(f(x), x) > 3\varepsilon$. One can refine'' the CW-structure to a new one such that every cell has diameter $< \varepsilon$. By cellular approximation we can see that there exists a cellular map $g \colon X \to X$ with $g \simeq f$ and $d(g(x),f(x)) < \varepsilon$. Thus $g(\overline e) \cap \overline e = \emptyset$ for every cell $e$. Hence, the diagonal matrix entries of each $C_i^{CW}(g)$ are zero and thus $\Lambda(g) = \Lambda(f) = 0$. \end{proof}  Jan-Bernhard Kordaß committed Oct 25, 2016 158   Jan-Bernhard Kordaß committed Oct 25, 2016 159   Jan-Bernhard Kordaß committed Nov 08, 2016 160 161 162 163 164 165 166 167 168 169 170 \section{Whitehead torsion} \subsection{Introduction/Motivation} Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-complexes, Whitehead torsion is an assignment $\tau(f) \in \Wh(\pi_1(Y))$ living in the so-called Whitehead group. \begin{thm*}[Properties of Whitehead torsion] \begin{enumerate}[label=(\arabic*)] \item homotopy invariance \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$. \item additivity:  Jan-Bernhard Kordaß committed Nov 23, 2016 171  A cellular push-out is a diagram  Jan-Bernhard Kordaß committed Nov 08, 2016 172 173 174 175 176 177  \begin{equation*} \begin{tikzcd} X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_1 \ar{r} & X \end{tikzcd} \end{equation*}  Jan-Bernhard Kordaß committed Nov 23, 2016 178  with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a sub-complex.  Jan-Bernhard Kordaß committed Nov 08, 2016 179 180 181 182 183 184 185 186 187  If the diagram \begin{equation*} \begin{tikzcd} X_0 \ar{rr} \ar[hook]{dd} \ar{rd}{f_0}[swap]{\simeq} &[0.8cm] &[-0.2cm] X_2 \ar[bend left=10]{rd}{f_2}[swap]{\simeq} &[0.8cm] \\ & Y_0 \ar[near start]{rr}{\phi} \ar[near end,hook]{dd}{j} & & Y_2 \ar{dd}{i}\\ X_1 \ar[crossing over]{rr} \ar[bend right=10]{rd}{f_1}[swap]{\simeq} & & X \ar[dashed]{rd}{f} \ar[leftarrow,crossing over]{uu} & \\ & Y_1 \ar{rr}{\psi} & & Y \end{tikzcd} \end{equation*}  Jan-Bernhard Kordaß committed Nov 23, 2016 188  is a map of cellular push-outs such that $f_i$ are homotopy equivalences.  Jan-Bernhard Kordaß committed Nov 08, 2016 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444  Then $f$ is a homotopy equivalence and \begin{align*} \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\ & = \psi_{*}(\tau(f_1)) + j_{*}(\tau(f_2)) - (\psi \circ i)_{*}(\tau(f_0)) \in \Wh(\pi_1(Y)). \end{align*} A similar additivity holds for the Lefschetz number.\footnote{Idea. $0 \to C_1(X_0) \to C_{*}(X_1) \oplus C_{*}(X_2) \to C_{*}(X) \to 0$ exact.} \item composition formula. If we have $\begin{tikzcd} X \ar{r}{f}[swap]{\simeq} & Y \ar{r}{g}[swap]{\simeq} & Z \end{tikzcd}$, then \begin{align*} \tau(g \circ f) & = "\tau(f) + \tau(g)"\\ & = g_{*}(\tau(f) + \tau(g)) \in \Wh(\pi_1Z). \end{align*} \end{enumerate} \end{thm*} \begin{thm*}[s-cobordism theorem (Mazur, Barden, Stallings, Smale)] Let $M$ be a closed smooth manifold of dimension $\geq 5$. Let $(W, i, j)$ be an s-cobordism \begin{figure}[h!] \centering \begin{tikzpicture}[scale=0.8] \coordinate (L1) at (2,0); \coordinate (L2) at (2,3); \coordinate (LL1) at ($(L1)+(-4,0)$); \coordinate (LL2) at ($(L2)+(-4,0)$); \coordinate (R1) at (8,0.5); \coordinate (R2) at (8,2); \coordinate (RR1) at ($(R1)+(3,0)$); \coordinate (RR2) at ($(R2)+(3,0)$); % left inclusion \ellipsebetweenvert{LL1}{LL2} \node[below] at (LL1) {$M$}; \draw[right hook->] ($(LL1) + (1.2,1.5)$) to node[above] {$i$} node[below]{$\simeq$} ($(L1) + (-1.2,1.5)$); % cobordism \ellipsebetweenvert{L1}{L2} \node[below] at (L1) {$M_0$}; \draw[out=0,in=180] (L1) to (5,-1) to (R1); \draw[out=0,in=180] (L2) to (R2); \topgenus[0.35]{5,0} \topgenus[0.22]{6,1.2} \node at (6,3) {$W$}; \ellipsebetweenvert[left]{R1}{R2} \node[below] at (R1) {$M_1$}; % right inclusion \draw[left hook->] ($(RR1) + (-1,0.75)$) to node[above] {$j$} node[below]{$\simeq$} ($(R1) + (1,0.75)$); \ellipsebetweenvert{RR1}{RR2} \node[below] at (RR1) {$N$}; \end{tikzpicture} % sketch: \includegraphics[width=0.8\textwidth]{img/1.png} \end{figure} i.e. $\partial W = M_0 \coprod M_1$ and $i \colon M \hookrightarrow W$, $j \colon N \hookrightarrow W$ are homotopy equivalences. Then $\tau(M \xrightarrow{i} W) = 0$ if and only if $(W,i_0,i_1)$ is trivial, i.e. \begin{figure}[h!] \centering \begin{tikzpicture}[every node/.style={scale=0.8}] \coordinate (LU1) at (2.4,1.5); \coordinate (LU2) at (2.4,2.5); \coordinate (RU1) at (5,1.3); \coordinate (RU2) at (5,2.3); \coordinate (LD1) at (2.4,0.5); \coordinate (LD2) at (2.4,-0.5); \coordinate (RD1) at (5,0.5); \coordinate (RD2) at (5,-0.5); % leftmost part \node at (0,3) {$\exists$}; \node[scale=1.5] at (-1,1) {$M$}; \draw[right hook->] (-0.4,1.2) to node[above] {$i$} (2,2); \draw[right hook->] (-0.4,0.8) to node[below] {$\operatorname{incl}$} (2,0); % upper cobordism \ellipsebetweenvert{LU1}{LU2} \draw[out=0,in=180] (LU1) to (3,1.6) to (RU1); \draw[out=0,in=180] (LU2) to (4,2.2) to (RU2); \ellipsebetweenvert[left]{RU1}{RU2} % arrow from the cobordism to the cylinder \draw[->] ($(LU1)!0.5!(RU1) + (0,-0.1)$) to node[left]{$\cong$} node[right]{diffeo} ($(LU1)!0.5!(RU1) + (0,-0.7)$); % cylinder \ellipsebetweenvert{LD1}{LD2} \draw (LD1) to (RD1); \draw (LD2) to (RD2); \ellipsebetweenvert[left]{RD1}{RD2} \end{tikzpicture} % sketch: \includegraphics[width=0.5\textwidth]{img/2.png} \end{figure} \end{thm*} This theorem implies the Poincaré conjecture in dimensions at least 6, which says that if $M$ is a closed (smooth) manifold that is homotopy equivalent to $S^n$, then $M$ is homeomorphic to $S^n$. The proof of this implication is basically along these lines: Pick disjoint embedded $n$-disks in $M$ and remove them. The result is a manifold $W$ with two boundary components. Consider the s-cobordism theorem for this manifold as depicted in the figure below, where $f$ is a diffeomorphism of $(n-1)$-spheres. \begin{figure}[h!] \centering \begin{tikzpicture} \coordinate (U1) at (0,3); \coordinate (U2) at (2.2,3); \coordinate (D1) at (-0.2,0); \coordinate (D2) at (1.8,0); % bordism on the left \ellipsebetweenhor{U1}{U2} \draw[out=270,in=90] (U1) to (0.3,1.8) to (D1); \draw[out=270,in=90] (U2) to (2.3,1.7) to (D2); \ellipsebetweenhor[upper]{D1}{D2} \node[left] at (0,1.5) {$W$}; % arrows and lower disk \draw[bend left=15,->] (2.8,3.3) to node[above] {$f$} (4.2,3.3); \draw[->] (3,1.5) to node[above] {$\cong$} (4,1.5); \draw[left hook->] (2.5,-1) to (1.3,-0.5); \draw[right hook->] (4.2,-1) to (5.5,-0.5); \draw[pattern=north west lines,pattern color=gray] (3.35,-1) ellipse (0.6 and 0.3); \node at (3.8,-0.4) {$D^n$}; % cylinder on the right \ellipsebetweenhor{5,3}{7,3} \draw (5,3) to (5,0); \draw (7,3) to (7,0); \ellipsebetweenhor[upper]{5,0}{7,0} \node[right] at (7,1.5) {$S^{n-1} \times [0,1]$}; \end{tikzpicture} % sketch: \includegraphics[width=0.7\textwidth]{img/3.png} \end{figure} By filling top and bottom, the Poincaré conjecture is implied, if we can extend a diffeomorphism $f \colon S^{n-1} \xrightarrow{\cong} S^{n-1}$ to a homeomorphism $F \colon D^n \xrightarrow{\cong} D^n$. This can be done by the so-called Alexander trick ($F(t x) = tf(x)$ for $t \in [0,1], x\in S^{n-1}$). \subsection{Whitehead group and lower K-theory} Let $R$ be a unital ring. Then \begin{align*} K_0(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ isomorphism classes $[P]$ of fin. gen. projective $R$-modules $\}$ and relations $R = \{\ [P_1] = [P_0] + [P_2]$ whenever $0 \to P_0 \to P_1 \to P_2 \to 0$ is exact $\}$. (Recall that a direct summand of in a free $R$-module is called a \CmMark{projective module}.) This can be understood as a kind of universal dimension for projective $R$-modules. \begin{align*} K_1(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ conjugacy classes $[f]$ of automorphisms $f \colon P \to P$ of fin. gen. projective $R$-modules $\}$ and relations $R$ given as follows. \begin{enumerate}[label=(\roman*)] \item Every commuting diagram \begin{equation*} \begin{tikzcd} 0 \ar{r} & P_0 \ar{r} \ar{d}{f_0}[swap]{\cong} & P_1 \ar{r} \ar{d}{f_1}[swap]{\cong} & P_2 \ar{r} \ar{d}{f_2}[swap]{\cong} & 0\\ 0 \ar{r} & P_0 \ar{r} & P_1 \ar{r} & P_2 \ar{r} & 0. \end{tikzcd} \end{equation*} gives rise to a relation $[f_1] = [f_0] + [f_2]$. \item $f,g \colon P \xrightarrow{\cong} P$ yield a relation $[f \circ g] = [f] + [g]$. \end{enumerate} This can be understood as an attempt to define a universal determinant of an automorphism. There is a more common definition of $K_1(R)$ in terms of the general linear groups with coefficients in $R$. Recall that $\GL(R) = \colim_{n \to \infty} \GL_n(R)$ with respect to the inclusion $\GL_n(R) \hookrightarrow \GL_{n+1}(R)$ to the upper left block. Then \begin{align*} K_1(R) = \GL(R)_{\text{ab}} = \GL(R)/[\GL(R),\GL(R)]. \end{align*} The so-called \CmMark{Whitehead lemma} states that $[\GL(R),\GL(R)] = E(R)$, where $E(R)$ is the subgroup of $\GL(R)$ generated by all elementary upper triangular matrices with ones on the diagonal. As a consequence, if $R$ is a field then the determinant defines an isomorphism $\det \colon K_1(R) \xrightarrow{\cong} \R\setminus\{0\}$. To see the equivalence of these two definitions, we can use the map \begin{align*} \GL(R) & \to K_1(R)\\ A & \mapsto [R^n \to R^n, \ x \mapsto Ax] \end{align*} and the fact that is descents to $\GL(R)_{\text{ab}} \to K_1(R)$. The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \otimes \id} P \oplus Q \mid f \text{ iso } \}$. \begin{dfn*} Let $\Gamma$ be a group. Then the \CmMark{Whitehead group} is defined as \begin{align*} \Wh(\Gamma) := \coker(\Gamma \times \{\pm 1\} \to K_1(\Z[\Gamma]), \ (\gamma, \pm 1) \mapsto \pm[\gamma]) \end{align*} \end{dfn*} \begin{expl*} The Whitehead group $\Wh(\{1\})$ is trivial, since the determinant yields an isomorphism $K_1(\Z) \xrightarrow{\det} \{\pm 1\}$. It is a conjecture that torsion-free groups $\Gamma$ have vanishing Whitehead group. Assertion: $\Wh(\Z/5) \cong \Z$. Here only prove that $\Wh(\Z/5)$ is infinite. We have a map $\phi_{*} \colon K_1(\Z[\Z/5]) \to K_1(\C)$ induced by $\Z[\Z/5] \xrightarrow{\phi} \C, \ t \mapsto \xi$, where $\Z/5 \cong \left$ and $\xi = \exp(2\pi i/5) \in \C$. Thus \begin{equation*} \begin{tikzcd} K_1(\Z[\Z/5]) \ar{r}{\phi_{*}} \ar{d} & K_1(\C) \ar{r}{\det} & \C^{\times} \ar{r}{|\blank|} & \R_{> 0}\\ \Wh(\Z/5) \ar[bend right=10]{rrru}{\tau} & & & \end{tikzcd} \end{equation*} One can see that $1 - t - t^{-1}$ is a unit in $\Z[\Z/5]$, since $(1 - t - t^{-1})( - t^2 - t^3) = 1$ and thus $\tau([1 - t - t^{-1}]) \neq 1$ \end{expl*}  Jan-Bernhard Kordaß committed Nov 22, 2016 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 \section{Whitehead torsion for chain complexes} In the following let us repeat some preliminaries on chain complexes. Let $R$ be a (not necessarily commutative) ring. Let $f_{*} \colon C_{*} \to D_{*}$ be an $R$-chain map. The \CmMark{mapping cylinder} $\cyl(f_{*})$ is an $R$-chain complex with p-th differential \begin{align*} C_p \oplus C_{p-1} \oplus D_p \xrightarrow{% \begin{pmatrix} c_p & -\id & 0\\ 0 & -c_{p-1} & 0\\ 0 & f_{p-1} & f \end{pmatrix}} C_{p-1} \oplus C_{p-2} \oplus D_{p-1}, \quad \end{align*} \begin{rem*}  Jan-Bernhard Kordaß committed Nov 23, 2016 465  For a continuous map $f \colon X \to Y$ we have the topological mapping cylinder.  Jan-Bernhard Kordaß committed Nov 22, 2016 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550  \begin{equation*} \begin{tikzcd} X \ar{r}{f} \ar[hook]{d}{i_0} & Y \ar[hook]{d}\\ X \times [0,1] \ar{r} & \cyl(f) \end{tikzcd} \end{equation*} If $X,Y$ are CW-complexes and $f$ is cellular, then \begin{align*} \cyl(C_{*}(f) \colon C_{*}(X) \to C_{*}(Y)) = C_{*}(\cyl(f)). \end{align*} \end{rem*} The \CmMark{mapping cone} $\cone(f_{*} \colon C_{*} \to D_{*})$ is a quotient of $\cyl(f_{*})$ by the obvious copy of $C_{*}$, so its differential is \begin{align*} C_{p-1} \oplus D_+ \xrightarrow{% \begin{pmatrix} -c_{p-1} & 0 \\ f_{p-1} & d_p \end{pmatrix}} C_{p-2} \oplus D_{p-1} \end{align*} \begin{rem*} Again there is a topological analogue, the topological mapping cone $\cone(f) = \cyl(f)/X \times \{1\}$. These are related via \begin{align*} \cone_i(C_{*}(f)) = C_i(\cone(f)) \text{ for } i > 0. \end{align*} \end{rem*} The \CmMark{suspension} $\Sigma C_{*}$ of an $R$-chain complex $C_{*}$ is a chain complex with $p$-th differential \begin{align*} C_{p-1} \xrightarrow{-c_{p-1}} C_{p-2}, \end{align*} which is isomorphic to a quotient of $\cone(\id_{C_{*}})$ by $C_{*}$. We have two exact sequences \begin{align*} & 0 \to C_{*} \to \cyl(f_{*}) \to \cone(f_{*}) \to 0 & 0 \to D_{*} \to \cone(f_{*}) \to \Sigma C_{*} \to 0 \end{align*} \begin{dfn*} An $R$-chain complex $C_{*}$ is \CmMark{finite}, if $|C_p| = 0$ for $p >> 0$ and each $C_p$ is finitely generated. It is called \CmMark{projective}, if each $C_p$ is projective; \CmMark{free}, if each $C_p$ is free, and \CmMark{based free}, if each $C_p$ is based free with a preferred basis. \end{dfn*} \begin{rem*} Let $f_{*} \colon C_{*} \to D_{*}$ be a chain map between projective chain complexes. Then the following statements are equivalent \begin{enumerate} \item $f_{*}$ is a homology isomorphism (i.e. $H_i(f_{*})$ is an isomorphism for all $i$) \item $f_{*}$ is a chain homotopy equivalence \item $\cone(f_{*})$ is contractible (i.e. $\cone(f_{*}) \simeq 0$). \end{enumerate} This can be seen from the following sequence (together with the fundamental lemma in homological algebra to show the equivalence of the first two statements). \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_{*} \ar{r} & \cyl(f) \ar[<->]{d}{\simeq} \ar{r} & \cone(f_{*}) \ar{r} & 0\\ & & D_{*} & & \end{tikzcd} \end{equation*} A short exact sequence of chain complexes induces a long exact sequence in homology associated to $C_{*}$, which implies $\cone(f_{*}) = 0 \rightsquigarrow H_{*}(\cone(f_{*})) = 0\rightsquigarrow f_{*}$ is homology isomorphism. \end{rem*} \begin{lemma*} Let $C_+$ be a based free, finite $R$-chain complex that is contractible. Let $\gamma_p \colon C_p \to C_{p+1}$ for $p \in \Z$ be a chain contraction, i.e. \begin{align*} c_{p+1} \circ \gamma_p + \gamma_{p-1} \circ c_p = \id - 0. \end{align*} Then the $R$-homomorphism $(c_{*} + \gamma_{*}) \colon C_{\text{odd}} \to C_{\text{ev}}$ (where $C_{\text{odd}} = \bigoplus_p C_{2p+1}$ and $C_{\text{ev}} = \bigoplus_p C_{2p}$) is an isomorphism. Let $A$ be its representing matrix. Its class $[A] \in K_1(R)$ is independent of the choice of $\gamma_{*}$. \end{lemma*} \begin{expl*} Let $C_p = 0$ unless $i \in \{0,1,2\}$. Then \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_2 \ar{r}[swap]{c_2} & C_1 \ar{r}[swap]{c_1} \ar[bend right]{l}[swap]{\gamma_1} & C_0 \ar{r} \ar[bend right]{l}[swap]{\gamma_0} & 0 \end{tikzcd} \end{equation*}  Jan-Bernhard Kordaß committed Nov 23, 2016 551  is the full complex and thus contractible'' means short exact''.  Jan-Bernhard Kordaß committed Nov 22, 2016 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582  $C_1 \xrightarrow{\cong} C_0 \oplus C_2$ via $x \mapsto c_1 (x) + \gamma_1(x)$ with inverse $C_0 \oplus C_2 \xrightarrow{\cong} C_1, (x,y) \mapsto \gamma_0(x) + c_2(y)$. Let $\tilde\gamma$ be another chain contraction \begin{equation*} \begin{tikzcd} C_0 \oplus C_2 \ar{r}{\cong}[swap]{(\tilde \gamma_1, c_2)} & C_1 \ar{r}{\cong}[swap]{(c_1,\gamma_1)} & C_0 \oplus C_2 \end{tikzcd} \end{equation*} Let $x + y \in C_0 \oplus C_2$. Then \begin{align*} \tilde \gamma_0(x) + c_2(y) \mapsto \ & c_1\tilde \gamma_0(x) + \gamma_1 \tilde\gamma_0(x) + \gamma_1 c_2(y)\\ & = (x - \underbrace{\tilde\gamma_2c_0(x))}_{=0} + \gamma_1\tilde\gamma_0(x) + (y - \underbrace{c_3\gamma_2(y)}_{=0})\\ & = x + y + \gamma_1 \tilde\gamma_0(x), \end{align*} which is represented by a matrix $\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$. \end{expl*} \begin{proof} Let $\gamma_{*}, \delta_{*}$ be two chain contractions of $C_{*}$. Then we consider \begin{align*} (c_{*} + \delta_{*})_{\text{odd}} \colon C_{\text{odd}} \xrightarrow{A} C_{\text{ev}} \text{ and } (c_{*} + \delta_{*})_{\text{ev}} \colon C_{\text{ev}} \xrightarrow{B} C_{\text{odd}} \end{align*} represented by matrices $A$ and $B$. Define $\mu_n = (\gamma_{n+1} - \delta_{n+1}) \circ \delta_n$ and $\nu_n = (\delta_{n+1} - \gamma_{n+1}) \circ \gamma_n$. Then $(\id + \mu_{*})_{\text{odd}}$, $(\id + \nu_{*})_{\text{ev}}$ and $(c_{*} + \gamma_{*})_{\text{odd}} \circ (\id + \mu_{*})_{\text{odd}} \circ (c_{*} + \delta_{*})_{\text{ev}}$ are represented by upper triangular matrices with ones on the diagonal.  Jan-Bernhard Kordaß committed Nov 23, 2016 583  Thus $[A] = -[B]$ in $K_1(R)$ and $B$ is independent of the choice of $\gamma_{*}$, hence $A$ is independent of the choice of $\gamma_{*}$.  Jan-Bernhard Kordaß committed Nov 22, 2016 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 \end{proof} \begin{dfn*} \begin{enumerate}[label=(\roman*)] \item For a contractible, based free, finite $R$-chain complex $C_{*}$ define \begin{align*} \tau(C_{*}) := [(c_{*} + \gamma_{*})_{\text{odd}}] \in K_1(R) \end{align*} (for some or every) choice of $\gamma_{*} \colon C_{*} \simeq 0$). \item Let $f_{*} \colon C_{*} \to D_{*}$ be a chain homotopy equivalence of based free, finite $R$-chain complexes. The \CmMark{Whitehead torsion} of $f_{*}$ is \begin{align*} \tau(f_{*}) := \tau(\cone(f_{*})) \in K_1(R). \end{align*} \end{enumerate} \end{dfn*} We say that a short exact sequence of based free modules \begin{align*} 0 \to A \xrightarrow{j} B \xrightarrow{p} C \to 0 \end{align*} is \CmMark{based exact}, if $\text{basis}_B = B_1 \coprod B_2$, such that $B_1 = j(\text{basis}_{A)}$ and $p(B_2) = \text{basis}_C$. \begin{lemma*} Consider the following diagram with based exact rows. \begin{equation*} \begin{tikzcd} 0 \ar{r} & C_{*}' \ar{r} \ar{d}{f_{*}}[swap]{\simeq} & D_{*}' \ar{r} \ar{d}{g_{*}}[swap]{\simeq} & E_{*}' \ar{r} \ar{d}{h_{*}}[swap]{\simeq} & 0\\ 0 \ar{r} & C_{*} \ar{r} & D_{*} \ar{r} & E_{*} \ar{r} & 0 \end{tikzcd} \end{equation*} Then $\tau(g_{*}) = \tau(f_{*}) + \tau(h_{*})$. \end{lemma*} \begin{proof} We have the following diagrams with exact rows and columns. \begin{equation*} \begin{tikzcd} & 0 \ar{d} & 0 \ar{d} & 0 \ar{d} & \\ 0 \ar{r} & C_{*}' \ar{r} \ar{d}{\simeq} & D_{*}' \ar{r} \ar{d}{\simeq} & E_{*}' \ar{r} \ar{d}{\simeq} & 0\\ 0 \ar{r} & \cyl(f_{*}) \ar{r} \ar{d} & \cyl(g_{*}) \ar{r} \ar{d} & \cyl(h_{*}) \ar{r} \ar{d} & 0\\ 0 \ar{r} & \cone(f_{*}) \ar{r} \ar{d} & \cone(g_{*}) \ar{r} \ar{d} & \cone(h_{*}) \ar{r} \ar{d} & 0\\ & 0 & 0 & 0 \end{tikzcd} \end{equation*} We may assume that the short exact sequence given by the columns \begin{align}\label{eq:ses-1} 0 \to C_{*} \xrightarrow{j_{*}} D_{*} \xrightarrow{p_{*}} E_{*} \to 0 \end{align} is a based exact sequence of contractible based exact sequence of contractible based free, finite chain complexes and then have to prove that $\tau(D_{*}) = \tau(C_{*}) + \tau(E_{*})$. The sequence \eqref{eq:ses-1} splits as chain complexes.\footnote{This heavily depends on contractibility and is not true for arbitrary chain complexes.} Let $e_*$ be a contraction of $E_{*}$ and let $\sigma_i \colon E_i \to D_i$ be a split of $p_i$ for all $i$. Set $s_i \colon E_i \to D_i, s_i := d_{i+1} \circ \sigma_{i+1} \circ \varepsilon_i + \sigma_i \circ \varepsilon_{i-1} \circ e_i$. Claim: $s_{*}$ is a chain map and $p_{*} \circ s_{*} = \id_{E_{*}}$. Hence we obtain an isomorphism of chain complexes \begin{align*} (j_{*},s_{*}) \colon C_{*} \oplus E_{*} \xrightarrow{\cong} D_{*}, \end{align*} which has a corresponding matrix of the form $\begin{pmatrix}\Id & * \\ 0 & \Id \end{pmatrix}$. We can finish the proof with the following remark. If $u_{*} \colon C_{*} \to D_{*}$ is a chain isomorphism of then \begin{align*} \tau(C_{*}) - \tau(D_{*}) = \sum_p (-1)^p [u_p] \in K_1(R). \end{align*} This can be shown by transporting a chain contraction $\gamma_{*}$ for $C_{*}$ to one for $D_{*}$ via $u_{*}$, which yields a diagram: \begin{equation*} \begin{tikzcd} C_{\text{odd}} \ar{r} \ar{d}{u_{\text{odd}}}[swap]{\simeq} & C_{\text{ev}} \ar{d}{u_{\text{ev}}}[swap]{\simeq}\\ D_{\text{odd}} \ar{r} & D_{\text{ev}}. \end{tikzcd} \end{equation*} \end{proof}  Jan-Bernhard Kordaß committed Oct 25, 2016 661 662 663  \chapter{Harmonic Maps [Andy Sanders]}  Jan-Bernhard Kordaß committed Nov 08, 2016 664 Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}.  Jan-Bernhard Kordaß committed Nov 08, 2016 665   Jan-Bernhard Kordaß committed Oct 25, 2016 666 667 \section{Basics of harmonic maps}  Jan-Bernhard Kordaß committed Nov 08, 2016 668 669 In the following let every manifold be oriented (for integration safety reasons).  Jan-Bernhard Kordaß committed Oct 25, 2016 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 \subsection{Background differential geometry} Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$. A \CmMark{connection} $\nabla$ on $E$ is an $\R$-linear map \begin{align*} \nabla \colon \Omega^0(E) \to \Omega^0(\T^{*}M \otimes_{\R} E) =: \Omega^1(M,E), s \mapsto \nabla_{\blank} s \end{align*} where $\Omega^0(E)$ denotes smooth sections in $E$, such that \begin{enumerate} \item $\nabla_{X+Y}s = \nabla_Xs + \nabla_Ys$, \item $\nabla_X(s+s') = \nabla_X s + \nabla_Xs'$ \item $\nabla_{fX} s = f\nabla_Xs$ \item $\nabla_X(fs) = f\nabla_Xs + X(f) s$. \end{enumerate} Let $q$ be an inner product on $E$. We say that $\nabla$ is a \CmMark{metric connection} for $q$, if for all $s,t \in \Omega^0(E)$ we have \begin{align*}  Jan-Bernhard Kordaß committed Nov 08, 2016 689  \dop q(s,t) = q(\nabla s,t) + q(s, \nabla t).  Jan-Bernhard Kordaß committed Oct 25, 2016 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 \end{align*} \begin{expl*} Let $(M,g)$ be a riemannian manifold with tangent bundle $E = \T M$ and Levi-Civita connection $\nabla$ of $g$. Let $X,Y \in \Omega^0(M)$ be vector fields, i.e. $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$ in local co-ordinates. (Abbreviate $\partial_i$ for $\frac{\partial}{\partial x^i}$.) \begin{align*} \nabla_XY = \nabla_{X^i\partial_i} Y^i\partial_i = X^i(\nabla_{\partial_i}Y^i\partial_i) = X^i(\partial_iY^i\partial_i + Y^i\nabla_{\partial_i}\partial_i) = X^i(\partial_iY^i\partial_i + Y^i\Gamma_{ij}^k\partial_i) \end{align*} where $\Gamma_{ij}^k = g^{km}(\partial_ig_{im} + \partial_j g_{im} - \partial_mg_{ij})$ for $g_{ij} = g(\partial_i,\partial_j)$ and $g^{km}$ is the $km$-entry of $g^{-1}$. \end{expl*} Out of $E$ one can build another bundle $E^{*} = \Hom(E,\R)$ and given another vector bundle $F$, one can build $\Hom(E,F)$, \begin{dfn*} Let $(E,\nabla) \to M$ be a vector bundle with a connection over $M$. The space of \CmMark{$p$-forms} on $m$ with values in $E$ is the $C^{\infty}(M)$-module $\Omega^p(M,E) = \Omega^0(M,\bigwedge^p\T^{*}M \otimes E)$. Elements $\alpha$ in $\Omega^p(M,E)$ have representations as linear combination of $\alpha_{i_1,\cdots,i_p}\dop x^{i_1} \wedge \cdots \wedge \dop x^{i_p} \otimes (s_1, \cdots s_p)$. \end{dfn*} \begin{dfn*} The exterior covariant derivative is the map given by extension of \begin{align*} \dop^{\nabla} \colon \Omega^p(M,E) & \to \Omega^{p+1}(M,E),\\ \alpha \otimes u & \mapsto \dop^{\nabla}(\alpha \otimes u) = \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u \end{align*} for $\alpha \in \bigwedge^p\T^{*}M$, $u \in \Omega^0(E)$. \end{dfn*} We want to define an inner product on $\Omega^p(M,E)$. For this, fix a metric $g$ on $M$ and let $(E,\nabla,q) \to M$ be a vector bundle with metric and connection over $M$. \begin{align*} \left< \alpha \otimes u, p \otimes v\right> = \int_M g(\alpha,p) q(u,v) \dop v_g \end{align*} is a number. (For this integral to be finite, assume $M$ is compact or work with compactly supported sections.) \begin{dfn*} The \CmMark{exterior covariant codifferential}\footnote{non-standard notation} is the formal $L^2$-adjoint of $d$ \begin{align*} \delta^{\nabla} \colon \Omega^p(M,E) \to \Omega^p(M,E) \end{align*} such that $\left< \dop^{\nabla}(\alpha \otimes u), \beta \otimes v\right> = \left<\alpha \otimes u, \delta^{\nabla}(\beta \otimes v)\right>$. \end{dfn*} \begin{rem*}[Fact]  Jan-Bernhard Kordaß committed Nov 23, 2016 741  An integration by parts argument shows that $\delta^{\nabla}$ exists and, when $\nabla$ is a metric connection, then  Jan-Bernhard Kordaß committed Oct 25, 2016 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817  \begin{align*} \delta^{\nabla} \colon \Omega^1(M,E) \to \Omega^0(M,E), \ \alpha \otimes u \mapsto -\tr_g(\nabla^{\T^{*} \otimes E} \alpha \otimes u), \end{align*} where for $\Omega^1(M,E) \to \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$, we can take a trace with the metric by choosing an orthonormal basis. \end{rem*} \begin{dfn*} A \CmMark{harmonic $p$-form} with values in $E$ is an element $\omega_i \in \Omega^p(M,E)$ such that $\delta^{\nabla} = \delta^{\nabla} \omega = 0$. As a matter of fact this is equivalent to $\Delta \omega = 0$ for $\Delta := \delta^{\nabla} \circ \dop^{\nabla} + \dop^{\nabla} \circ \delta^{\nabla}$ (Consider $\left<\Delta \omega, \omega\right>$ and utilize the obvious stuff). \end{dfn*} \subsection{Definition of harmonic maps of 1st variation formula} Let $(M,g)$ and $(N,h)$ be two riemannian manifolds and let $f \colon M \to N$ be a smooth map. Then $\dop f \colon \T M \to \T N$ is an element $\dop f \in \Omega^0(\Hom(\T M, f^{*}\T N)) = \Omega^0(\T^{*}M \otimes f^{*}\T N)$. Next, the metrics $g,h$ induce a metric on $\T^{*}M \otimes f^{*}\T N$. \begin{dfn*} The energy density of $f \colon M \to N$ is $e(f) := \frac{1}{2} \left< \dop f, \dop f\right>_{\T^{*}M \otimes f^{*}\T N} = \frac{1}{2} \|\dop f\|^2$. \end{dfn*} Choose co-ordinates $\{x^i\}$ in $M$ and $\{y^i\}$ in $N$. With respect to these, we have \begin{align*} \frac{1}{2} \|\dop f \|^2 = \frac{1}{2}y^{ij} \partial_if^{*}\partial_jf^{\beta}h_{\alpha\beta}(f). \end{align*} \begin{dfn*} The \CmMark{Dirlichlet energy} is given by \begin{align*} E \colon C^2_0(M,N) \to \R, \ f \mapsto \int_M e(f) \dop V_g. \end{align*} A \CmMark{critical map} (or \CmMark{stationary map}) is a map $f \colon M \to N$ such that for all compactly supported $F \colon M \times (-\varepsilon, \varepsilon) \to N$ $C^2$-map (variation of $f$) with $F(x,0) = f(x)$ we have that \begin{align}\label{eq:first-variation} \delta E(\nu) := \left.\frac{\dop}{\dop t} E(F) \right|_{t = 0} = 0 \end{align} for $\nu = \frac{\dop}{\dop t} F|_{t = 0} \in \Omega^0(f^{*}\T N)$. The \cref{eq:first-variation} is called \CmMark{first variation in the direction of $\nu$}. \end{dfn*} \begin{dfn*} The map $f \colon (M,g) \to (N,h)$ is called \CmMark{harmonic}, if it is a critical point for the Dirlichlet energy. \end{dfn*} \begin{dfn*} Let $\dop f \in \Omega^1(M, f^{*}\T N)$ then $\nabla \dop f \in \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$. The \CmMark{second fundamental form} of $f$ is $\nabla \dop f := B_f$, which is a symmetric $2$-tensor on $M$. \end{dfn*} \begin{dfn*} The \CmMark{tension field} of $f$ is the trace of $B_f$: $\tau(f) := \tr_g(B_f) \in \Omega^0(M,f^{*}\T N)$. \end{dfn*} \begin{thm*}[1st variation of $E$] Let $F \colon M \times (\varepsilon, \varepsilon) \to N$ a variation of $f$ and let $\nu = \frac{\dop}{\dop t}F|_{t = 0}$. Then \begin{align*} \delta E(\nu) = \frac{\dop}{\dop t}E(F)|_{t = 0} = - \int_M \left<\tau(f), \nu\right> \dop v_g. \end{align*} \end{thm*} \begin{proof} The variation $F \colon M \times (-\varepsilon,\varepsilon) \to N$ yields a pullback connection on $F^{*}\T N$, which shows \begin{align*} \frac{\dop}{\dop t}E(F)|_{t = 0} & = \frac{1}{2} \int_M \frac{\dop}{\dop t}\left<\dop F, \dop F\right> \dop V_g|_{t = 0} = \int_M \left<\nabla_{\frac{\partial}{\partial t}}\dop F, \dop F\right> \dop V_g|_{t = 0}\\ & = \int_M \left<\nabla^{f^{*}\T N}\nu, \dop f\right> \dop V_g \overset{(*)}{=} \int_M \left<\nu, \delta^{\nabla^{f^{*}\T N}} \dop f\right> \dop V_g\\ & = -\int_M \left< \nu, \tr_g(\nabla \dop f) \right> \dop V_g  Jan-Bernhard Kordaß committed Nov 08, 2016 818 819 820  = - \int_M \left< \nu, \tau(f)\right> \dop V_g, \end{align*} where $(*)$ follows by a calculation in local co-ordinates.  Jan-Bernhard Kordaß committed Oct 25, 2016 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 \end{proof} \begin{cor*}[Fundamental theorem of the calculus of variations] A $C^2$-map $f \colon (M,g) \to (N,h)$ is harmonic if and only if $\tau(f) = 0$. \end{cor*} What does $\tau(f) = 0$ look like? Fix local co-ordinates $\{x^i\}$ on $M$ and $\{y^j\}$ on $N$. Then $\dop f = \partial_if^{alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}$ and thus \begin{align*} \nabla \dop f & = \nabla \partial_i f^{\alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} = \partial_j\partial_i f^{\alpha} \dop x^j \otimes \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} + \partial_if^{\alpha} \nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}\\ & = A + \partial_i f^{\alpha}(\nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \otimes \nabla \frac{\partial}{\partial y^{\alpha}})\\ & = A + \partial_i f^{\alpha}( -\Gamma^i_{jk} \dop x^i \otimes \dop x^k \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \partial_j f^{\beta} \Gamma^{\gamma}_{\alpha\beta} \frac{\partial}{\partial y^{\gamma}})\\ & = \partial_i \partial_jf^{\gamma} \Gamma_{ij}^k \partial_k f^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_jf^{\alpha}\partial_if^{\beta)} \dop x^i \otimes \dop x^j \otimes \frac{\partial}{\partial y^j}. \end{align*} Thus $\tau(f) = (\Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})$.  Jan-Bernhard Kordaß committed Nov 08, 2016 843 844 845 846 847 848 \section{Example and the Bochner formula (a glimpse of rigidity)} Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tension field \begin{align*} \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \end{align*}  Jan-Bernhard Kordaß committed Nov 23, 2016 849 In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and Greek ones for $N$.}  Jan-Bernhard Kordaß committed Nov 08, 2016 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 \begin{align*} \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0, \end{align*} where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \begin{expl*} \begin{enumerate}[label=\Roman*.] \item Let $(M,g) = (\R, \dop t^2)$ and let $\eta \colon \R \to (N,h)$. From above we know that the Laplace-Beltrami operator here reads \begin{align*} \Delta_gf = g^{ij} (\partial_i\partial_jf - \Gamma_{ij}^k\partial_kf) = g^{ij}(\partial_i\partial_jf) = \partial_t^2f \end{align*} for \begin{align*} \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij}) \end{align*}  Jan-Bernhard Kordaß committed Nov 22, 2016 868  ($=0$ if $\{g_{ij}\}$ is constant) and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$.  Jan-Bernhard Kordaß committed Nov 08, 2016 869 870 871 872  Hence \begin{align*} \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \end{align*}  Jan-Bernhard Kordaß committed Nov 23, 2016 873  which is if and only if $\eta$ is a geodesic, i.e. the covariant derivative along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$.  Jan-Bernhard Kordaß committed Nov 08, 2016 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 \item Now let $f \colon (M,g) \to \R$. Here $\tau(f) = \Delta_gf = 0$. \begin{prop*} If $M$ is closed, then the energy harmonic functions are constant. \end{prop*} \begin{proof} By Green's theorem (integration by parts) \begin{align*} \int_M \underbrace{g(\nabla f, \nabla f)}_{=\|\nabla f\|^2} \dop V_g = - \int \Delta_g f \cdot f \dop V_g = 0 \end{align*} for $\dop V_g = \sqrt{\det (\{g_{ij}\})} \dop x^1 \wedge \cdots \wedge \dop x^n$. Thus $\|\nabla f\|^2 = 0$ and $f$ must be constant. \end{proof} In our example, this shows $\Delta_g f = \lambda f$. \item Let $f \colon (M,g) \to (N,h)$ be an isometric immersion, i.e. $\dop f$ is injective and $g = f^{*}h = h(\dop f, \dop f)$. Then we have \begin{align*} e(f) & = \frac{1}{2} \|\dop f\|^2 = \frac{1}{2}h_{\alpha\beta} \partial_if^{\alpha}\partial_jf^{\beta}g^{ij} = \frac{1}{2}\partial_if^{\alpha}\partial_jf^{\beta}h(\partial_{\alpha},\partial_{\beta}) g^{ij}\\ & = \frac{1}{2}h(\partial_if^{\alpha}\partial_{\alpha},\partial_jf^{\beta}\partial_{\beta})g^{ij} = \frac{1}{2}h(\dop f(\partial_i),\dop f(\partial_j)) g^{ij} = \frac{1}{2}g_{ij}g^{ij} = \frac{m}{2} \end{align*} and hence $E(f) = \frac{m}{2}\Vol(f)$, where $\Vol(f) = \int_M\dop V_{f^{*}h} = \int_M\dop V_g$. This shows that $f$ is critical for $E$ if and only if $f$ is critical for $\Vol \colon \Imm(M,N) \to \R_+$. The latter is clearly if and only if $f$ is a \textbf{minimal submanifold}. Examples of minimal submanifolds in $\R^3$ include the 2-plane, or the helicoid. \end{enumerate} \end{expl*} \subsection{Composition laws for harmonic maps} Consider the composition \begin{align*} (M,g) \xrightarrow{f} (N,h) \xrightarrow{u} (Z,b). \end{align*} In general, if $f,u$ are harmonic, this needs not be harmonic again, which can be considered a bug or a feature''. \begin{align*} B_{u \circ f}(X,Y) = B_u(\dop f(X), \dop f(Y)) + \dop u (B_f(X,Y)) \end{align*} for $X,Y \in \T_pM$ and thus $B_{u \circ f} = \nabla^{\T^{*}M \otimes (u \circ f)^{*}\T N}(\dop(u \circ f))$. Hence $\tau(u \circ f) = \dop (\tau(f)) + \tr_g(f^{*}B_u)$. If $f$ is harmonic, then $\tau(u \circ f) = \tr_g(f^{*}B_u)$. \begin{prop*} If $f \colon M \to N$, is harmonic and $u \colon N \to Z$ is totally geodesic, i.e. $B_u = 0$. Then $u \circ f$ is harmonic. \end{prop*} What if $u \colon N \to \R$ is a function and $f$ is harmonic? Then \begin{align*} \tau(u \circ f) = \tr_g(f^{*}B_u) = \tr_g(f^{*}(\Hess(u)) = \sum_{i = 1}^n f^{*}(\Hess(u)) (E_i,E_i). \end{align*} Recall that a function $u \colon (N,h) \to \R$ is convex, if $\Hess(u)$ is positive definite. If $f$ is harmonic and $u$ is convex, then $\tau(u \circ f) = \nabla_g u \circ f \geq 0$ (these are called \CmMark{subharmonic functions}). \begin{thm*} A map is harmonic if and only if it pulls back germs of convex functions to germs of subharmonic functions. \end{thm*} There are various useful applications of the synthetic view'' on harmonic functions (e.g. Gromov-Shane). \begin{thm*} Suppose $(M,g)$ is closed, connected and $(N,h)$ is $1$-connected with non-positive curvature. Then every harmonic map $f \colon (M,g) \to (N,h)$ is constant. \end{thm*} \begin{proof} The distance function $N \to \R_{\geq 0}, x \mapsto \dop_N(p,x)^2$ for every $p \in N$ is actually smooth and strictly convex, e.g. $\dop_{\R^n}(0,x)^2 = x_1^2 + \cdots + x_n^2$. In case $f$ is harmonic, we have \begin{align*} \Delta_gu \circ f = \tau(u \circ f) = \tr_g(f^{*}B_u) \geq 0 \end{align*} and \begin{align*} -\int \| \dop(u \circ f)\|^2 \dop V_g = \int_M \Delta_gu \circ f \dop V_g \geq 0. \end{align*} Thus $\|\dop (u \circ f)\| = 0$ and hence $u \circ f$ is constant. \end{proof} \subsection{Bochner formulas}  Jan-Bernhard Kordaß committed Nov 23, 2016 964 Let $(E,\nabla,a)$ be a riemannian vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$.  Jan-Bernhard Kordaß committed Nov 08, 2016 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023  Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$. \begin{align*} \hat\nabla \colon \Omega^p(M,E) \to \Omega^p(M,\T^{*}M \otimes \T^{*}M \otimes E), \quad \omega \mapsto ( (X,Y) \mapsto \nabla_X\nabla_Y\omega - \nabla_{\nabla_XY}\omega ) \end{align*} The \CmMark{trace Laplacian} is the operator \begin{align*} \nabla^2 \colon \Omega^p(M,E) \to \Omega^p(M,E), \quad \omega \mapsto \tr_g(\hat\nabla \omega). \end{align*} Recall that the \CmMark{Hodge Laplacian} was the operator \begin{align*} \dop^{\nabla} \colon \Omega^p(M,E) \to \Omega^{p+1}(M,E), \quad \alpha \otimes u \mapsto \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u. \end{align*} With respect to the $L^2$-pairing $\beta \otimes v \mapsto \int_Mg(\alpha,\beta) a(u,v)\dop V_g$ it has a formal adjoint \begin{align*} \delta^{\nabla}\colon \Omega^{p+1}(M,E) \to \Omega^p(M,E). \end{align*} The Hodge Laplacian is the degree preserving operator given by $\dop^{\nabla} \circ \delta^{\nabla} + \delta^{\nabla} \circ \dop^{\nabla} =: \Delta_a$. The \CmMark[Bochner-Lichnerowicz formula]{(generalized) Bochner-Lichnerowicz formula} is given by \begin{align*} \nabla_a \omega = - \nabla^2\omega + S_{\omega}. \end{align*} for $S_{\omega} \in \Omega^p(M,E)$ with \begin{align*} S_{\omega}(X_1, \cdots, X_p) = \sum_{k = 1}^p\sum_{i = 1}^m(-1)^k(R^{\tilde \nabla}(e_i, X_k)\omega) (e_i,X_1, \ldots, \hat X_k, \ldots, X_n) \end{align*} for $X_i \in \T_pM$, $m = \dim M$ and $\{e_i\}$ an orthonormal frame around $p$.\footnote{Hat ($\hat X_k$), as always, means to omit the k-th term.} \begin{cor*} Let $f \colon (M,g) \to (N,h)$ be harmonic. Then \begin{align*} \Delta_ge(f) = \|B_f\|^2 - \sum_{ij}\underbrace{h(R^h(f_{*}e_i,f_{*}e_j) f_{*}e_j, f_{*}e_i))}_{= \lambda \sec(e_i,e_j)} + \sum_ih(f_{*}(\Ric^g(e_i)),f_ke_i) \end{align*} for an orthonormal frame $\{e_i\}$. \end{cor*} The key observation for an application of this is that, if $\Ric^g$ is a positive operator, then the latter sum is positive. \begin{thm*}[Eells-Sampson] Let $(M,g)$ be a closed with non-negative Ricci curvature and let $(N,h)$ have non-positive sectional curvature. \begin{enumerate}[label=(\roman*)] \item Then any harmonic map $f \colon (M,g) \to (N,h)$ is totally geodesic, i.e. $\nabla \dop f = B_f = 0$. \item If $\Ric^g$ is positive at any point, then $f$ is constant. \item If the sectional curvature of $(N,h)$ is strictly negative, then $f$ is constant or $f(M)$ is closed geodesic. \end{enumerate} \end{thm*} \begin{proof} The first statement easily follows from the corollary and $\int_M \left< \nabla u, \nabla v\right> \dop V_g = - \int_M \Delta u \cdot v \dop V_g$. The second is also not that hard and the last requires some work. \end{proof}  Jan-Bernhard Kordaß committed Nov 22, 2016 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039  \section{The Eells-Sampson existence theorem} \textbf{Story:} Given two manifolds $M,N$, is there a best map in a given free homotopy class $\beta \in [M,N]$, where $[M,N]$ denotes the free homotopy classes of smooth maps. From now on, best'' means harmonic with respect to some riemannian metric. \begin{expl*} If $M = S^n$, then it is a theorem that every homotopy class $\gamma \in [S^1,N]$ (for $N$ closed) admits a harmonic representative $\gamma \colon S^1 \to (N,h)$, i.e. is a closed geodesic. \end{expl*} \begin{expl*} What about $\dim(M) \geq 2$. In this case it depends on the curvature of $(N,h)$. Consider the flat torus $\mathbb T^2$ and the round sphere $\mathbb S^2$.  Jan-Bernhard Kordaß committed Nov 23, 2016 1040  For a degree 1 map $\mathbb T^2 \to \mathbb S^2$ there is no harmonic map in the homotopy class (see the book by Lin on geometry of harmonic maps).  Jan-Bernhard Kordaß committed Nov 22, 2016 1041 1042 1043 1044 1045 1046 1047 1048 1049 \end{expl*} \begin{thm*}[Eells-Sampson 1964] Let $(M,g), (N,h)$ be closed manifolds and $h$ with non-positive sectional curvature. Then given any $f \colon M \to N$ $C^2$-map there exists a harmonic map $u \colon (M,g) \to (N,h)$ such that $u$ is freely homotopic to $f$. \end{thm*} Try to take $\tau(u) = 0$ for some $u \sim f$.  Jan-Bernhard Kordaß committed Nov 23, 2016 1050 In this approach $E \colon C^2(M,N) \to \R, f \mapsto \frac{1}{2} \int_M \|\dop f\|^2 dV_g$ such that $E(f_n) \to \inf_{f \in C^2}E(f)$, we would have to weaken to topology considering Sobolev spaces $W^{1,2}(M,N)$  Jan-Bernhard Kordaß committed Nov 22, 2016 1051 1052  The other approach using gradient flow goes as follows.  Jan-Bernhard Kordaß committed Nov 23, 2016 1053 Try to solve initial value problem (IVP).  Jan-Bernhard Kordaß committed Nov 22, 2016 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 Let $f \colon M \times (0,\infty) \to N$, such that $\frac{\partial f}{\partial t} = \tau(f_t)$ and $f(\blank, 0) = f$. Recall the first variation of every $f_t \colon M \to N, \frac{\dop}{\dop t}f_t|_{t = 0} = 0$. Then $\delta E(\nu) = \frac{\dop}{\dop t}E(f_t)|_{t=0} = -\int_M \left<\tau(f),\nu\right>dV_g = -Q(\tau(f),\nu)$, where $\left<\blank,\blank\right>$ is the inner product on $f^{*}\T N$ induced by $h$. If we manage to solve $\frac{\partial f}{\partial t} = \tau(f)$, then \begin{align*} \frac{\dop}{\dop t} E(f_t)|_{t = t_0} = \int_M \left<\tau(f),\tau(f_t) \right> \dop V_g \leq 0 \end{align*} and equal to zero if and only if $\tau(f_{t_0}) = 0$. $\frac{\partial f^{\gamma}}{\partial t} = \Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f)\partial_if^{\alpha}\partial_if^{\beta}g^{il}$. \subsection{1st short time existence} \begin{thm*} Suppose $f \colon M \to N$ is a $C^2$-map. Then there exists a $T_{\text{max}} > 0$ such that (IVP) \begin{align*} \frac{\partial f_t}{\partial t} = \tau(f_t) \text{ and } f_0 \equiv f \end{align*} has a solution on $[0, T_{\text{max}}]$. If $T_{\text{max}} < \infty$, then \begin{align*} \limsup_{t \nearrow T, x \in M}(f_t) = + \infty. \end{align*} \end{thm*} Note that there is no assumption on the curvature. \subsection{Need another Bochner formula} Let $(N,h)$ has non-positive sectional curvature and let $M$ be an $m$-dimensional manifold. Then we can calculate \begin{align*} & \frac{\partial}{\partial t} e(f_t) - \Delta_ge(f_t)\\ & \quad = -\underbrace{\|B_{f_t}\|^2}_{=\nabla \dop f_t} - \sum_{i=1}^n h(\sum_{j = 1}^m \dop f_t(\Ric^g(e_{i},e_j)e_j),\dop f_t(e_i))\\ & \qquad + \underbrace{\sum_{i,j = 1}^m h(R^h(\dop f_t(e_i), \dop f_t(e_j))\dop f_t(e_j),\dop f_t(e_i))}_{ \leq m}, \end{align*} where $\Ric^g \colon \T M \otimes \T M \to \R$ is the Ricci tensor, $R^h$ is the full curvature tensor of $(N,h)$ and $\{e_1, \ldots, e_m\}$ is an orthonormal frame of $N$. The latter summand is $\operatorname{const} \sec^h(\operatorname{span}(\dop f_t(e_i),\dop f_t(e_j)))$.  Jan-Bernhard Kordaß committed Nov 22, 2016 1101 We continue\footnote{TODO: According to Andy, who erased this part very quickly, there should be some mistake somewhere here...}  Jan-Bernhard Kordaß committed Nov 22, 2016 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 \begin{align*} \leq -\sum_{i = 1}^m h(\sum_{j=1}^m \dop f_t(\Ric^g(e_i,e_j)e_j),\dop f_t(e_i)) \leq C \sum_{i,j = 1}^m h(\dop f_t(e_i, \dop f_t(e_j)) \leq C e(f_t). \end{align*} Thus we get the following theorem. \begin{thm*} If $(N,h)$ has non-positive sectional curvature, then \begin{align*} \frac{\partial}{\partial t}e(f_t) - \Delta_ge(f_t) \leq C e(f_t). \end{align*} \end{thm*} \subsection{Moser-Harnack inequality} Let $z_0 = (x_0,t_0) \in M \times (0,T)$ and let $0 < R < \min\{ \text{inf radius of } g , t_0 \}$. The parabolic cylinder is given as follows \begin{align*} P_R(z_0) = \{ z = (x,t) \in M \times (0,\infty) \mid \dop g(x,x_0) < R, t_0 - R^2 \leq t \leq t_0 \}. \end{align*} \begin{thm*}[Moser] Suppose $v \in C^2(P_R(z_0))$ is non-negative and satisfies \begin{align*} \frac{\partial v}{\partial t} - \Delta_gv \leq Cv \text{ for } C > 0. \end{align*} Then there exists a $C_1 > 0$ such that \begin{align*} v(z_0) \leq C_1R^{2-m}\int_{P_R(z_0)}v \dop V_g \dop t. \end{align*} \end{thm*} If we apply this, we obtain \begin{align*} e(f_t)(z_0) \leq CR^{2-n} \int_{}\int_{} e(f_t) \dop V_g \dop t \leq C R^{2-n} \int_{t_0 - R^2 }^{t_0} E(f_t) \dop t \leq C R^{2-n} E(f) R^2. \end{align*} If $T_{\text{max}} < \infty$, then recall $\limsup_{t \nearrow T, x \in M} e(f_t) = +\infty$, but we have proved $e(f_t)$ is uniformly bounded, hence $T_{\text{max}} = + \infty$. \subsection{Black box \# 37} Since $e(f_t)$ is bounded for all time, elliptic regularity'' implies that for all $m > 0$ we have $\|\nabla^m \dop f_t\| \leq C_m$. For $f \colon M \times [0,\infty) \to N$ by Arzela-Ascoli, we know that there exists a subsequence $t_k \to \infty$ such that $f(\blank, t_k) \to u$ for $t_k \to \infty$ (in the sense of $C^2$-convergence). We calculate \begin{align*} \int_0^{t_0} \int_M \big\|\frac{\partial f}{\partial t}\big\|^2 \dop V_g\dop t & = \int_0^{t_0} \int_M \|\tau(f_t)\|^2 \dop V_g \dop t = - \int_0^{t_0} \frac{\partial E}{\partial t}(f_t)\dop t\\ & = -E(f_t) + E(f) \leq E(f) < \infty. \end{align*} Hence $\limsup_{t_0 \nearrow +\infty} \int_{t_0-2}^{t_0}\int_M \|\frac{\partial f}{\partial t}\|^2\dop V_g\dop t = 0$. Now one computes a Bochner formula for $\|\frac{\partial f}{\partial t}\|^2_{C^0}$. This yields an equality of the following form. For each $0 < < 1$ we have and each $t > 0$ \begin{align*} \|\tau(f_t)\|^2_{C^{\alpha}(M \times [t-1,t])} = \big\|\frac{\partial f}{\partial t}\big\|^2_{C^{\alpha}(M \times [t-1,t])} < C(\alpha) \big\|\frac{\partial f}{\partial t}\big\|^2_{L^2(M \times [t-2,t])} \xrightarrow{t \to +\infty} 0. \end{align*} Hence there exists a subsequence $t_i$ such that \begin{align*} \| \tau(f_i)\|_{C^{\alpha}} \xrightarrow{t_i \to \infty} 0 \end{align*} and hence $0 = \lim \tau(f_i) = \tau(u)$. Since we have $C^2$-convergence, we can conclude that $u \sim f$.  Jan-Bernhard Kordaß committed Oct 25, 2016 1179 1180 1181 1182 %%% Local Variables: %%% mode: latex %%% TeX-master: "skript-rtg-lectures-ws1617" %%% End: