Riemannian formalism in potentially infinite dimensions

When one is talking of the canonical ensemble, one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature (thermostat). In this section we attempt to describe such an embedding using the formalism of Rimannian geometry.

6.1 Consider the manifold $\tilde{M}=M\times\mathbb{S}^N\times\mathbb{R}^+$ with the following metric:

$$\tilde{g}_{ij}=g_{ij}, \tilde{g}_{\alpha\beta}=\tau g_{\alpha\bet... ...=\frac{N}{2\tau}+R, \tilde{g}_{i\alpha}=\tilde{g}_{i 0}=\tilde{g}_{\alpha 0}=0,$$

where $i,j$ denote coordinate indices on the $M$ factor, $\alpha,\beta$ denote those on the $\mathbb{S}^N$ factor, and the coordinate $\tau$ on $\mathbb{R}^+$ has index 0; $g_{ij}$ evolves with $\tau$ by the backward Ricci flow $(g_{ij})_\tau=2R_{ij},$ $g_{\alpha\beta}$ is the metric on $\mathbb{S}^N$ of constant curvature $\frac{1}{2N}.$ It turns out that the components of the curvature tensor of this metric coincide (modulo $N^{-1}$) with the components of the matrix Harnack expression (and its traces), discovered by Hamilton [H 3]. One can also compute that all the components of the Ricci tensor are equal to zero (mod $N^{-1}$). The heat equation and the conjugate heat equation on $M$ can be interpreted via Laplace equation on $\tilde{M}$ for functions and volume forms respectively: $u$ satisfies the heat equation on $M$ iff $\tilde{u}$ (the extension of $u$ to $\tilde{M}$ constant along the $\mathbb{S}^N$ fibres) satisfies $\tilde{\triangle}\tilde{u}=0\$   mod$\ N^{-1};$ similarly, $u$ satisfies the conjugate heat equation on $M$ iff $\tilde{u}^*=\tau^{-\frac{N-1}{2}}\tilde{u}$ satisfies $\tilde{\triangle}\tilde{u}^*=0\$   mod$N^{-1}$ on $\tilde{M}.$

6.2 Starting from $\tilde{g},$ we can also construct a metric $g^m$ on $\tilde{M},$ isometric to $\tilde{g}$ (mod $N^{-1}$), which corresponds to the backward $m$-preserving Ricci flow ( given by equations (1.1) with $t$-derivatives replaced by minus $\tau$-derivatives, $dm=(4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$). To achieve this, first apply to $\tilde{g}$ a (small) diffeomorphism, mapping each point $(x^{i},y^{\alpha},\tau)$ into $(x^{i},y^{\alpha},\tau(1-\frac{2f}{N}));$ we would get a metric $\tilde{g}^m,$ with components (mod $N^{-1}$)

$$\tilde{g}^m_{ij}=\tilde{g}_{ij}, \tilde{g}^m_{\alpha\beta}=(1-\fr... ...\tilde{g}^m_{i 0}=-\nabla_i f, \tilde{g}^m_{i \alpha}=\tilde{g}^m_{\alpha 0}=0;$$

then apply a horizontal (that is, along the $M$ factor) diffeomorphism to get $g^m$ satisfying $(g^m_{ij})_\tau=2(R_{ij}+\nabla_i\nabla_j f);$ the other components of $g^m$ become (mod $N^{-1}$)

$$g^m_{\alpha\beta}=(1-\frac{2f}{N})\tilde{g}_{\alpha\beta}, g^m_{0... ...2=\frac{1}{\tau}(\frac{N}{2}-[\tau(2\triangle f-\vert\nabla f\vert^2 +R)+f-n]),$$

$$g^m_{i 0}=g^m_{\alpha 0}=g^m_{i \alpha}=0$$

Note that the hypersurface $\tau=$const in the metric $g^m$ has the volume form $\tau^{N/2}e^{-f}$ times the canonical form on $M$ and $\mathbb{S}^N,$ and the scalar curvature of this hypersurface is $\frac{1}{\tau}(\frac{N}{2}+\tau(2\triangle f-\vert\nabla f\vert^2+R)+f)$ mod $N^{-1}.$ Thus the entropy $S$ multiplied by the inverse temperature $\beta$ is essentially minus the total scalar curvature of this hypersurface.

6.3 Now we return to the metric $\tilde{g}$ and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem. Consider a metric ball in $(\tilde{M},\tilde{g})$ centered at some point $p$ where $\tau=0.$ Then clearly the shortest geodesic between $p$ and an arbitrary point $q$ is always orthogonal to the $\mathbb{S}^N$ fibre. The length of such curve $\gamma(\tau)$ can be computed as

$$\int_0^{\tau(q)}{\sqrt{\frac{N}{2\tau}+R+\vert\dot{\gamma}_M(\tau)\vert^2}d\tau}$$

$$=\sqrt{2N\tau(q)}+\frac{1}{\sqrt{2N}}\int_0^{\tau(q)}{\sqrt{\tau}(R+\vert\dot{\gamma}_M(\tau)\vert^2)d\tau}+ O(N^{-\frac{3}{2}})$$

Thus a shortest geodesic should minimize $\mathcal{L}(\gamma)=\int_0^{\tau(q)}{\sqrt{\tau}(R+\vert\dot{\gamma}_M(\tau)\vert^2)d\tau},$ an expression defined entirely in terms of $M$. Let $L(q_M)$ denote the corresponding infimum. It follows that a metric sphere in $\tilde{M}$ of radius $\sqrt{2N\tau(q)}$ centered at $p$ is $O(N^{-1})$-close to the hypersurface $\tau=\tau(q),$ and its volume can be computed as $V(\mathbb{S}^N)\int_M{(\sqrt{\tau(q)}-\frac{1}{2N}L(x)+O(N^{-2}))^Ndx},$ so the ratio of this volume to $\sqrt{2N\tau(q)}^{N+n}$ is just constant times $N^{-\frac{n}{2}}$ times

$$\int_M{\tau(q)^{-\frac{n}{2}}\mbox{exp}(-\frac{1}{\sqrt{2\tau(q)}}L(x))dx}+O(N^{-1})$$

The computation suggests that this integral, which we will call the reduced volume and denote by $\tilde{V}(\tau(q)),$ should be increasing as $\tau$ decreases. A rigorous proof of this monotonicity is given in the next section.

6.4* The first geometric interpretation of Hamilton's Harnack expressions was found by Chow and Chu [C-Chu 1,2]; they construct a potentially degenerate riemannian metric on $M\times \mathbb{R},$ which potentially satisfies the Ricci soliton equation; our construction is, in a certain sense, dual to theirs.

Our formula for the reduced volume resembles the expression in Huisken monotonicity formula for the mean curvature flow [Hu]; however, in our case the monotonicity is in the opposite direction.