1 Introduction

Given a closed (i.e. compact and without boundary) \(n\)-dimensional Riemannian manifold \(M\) with a one-parameter family of smooth Riemannian metrics \(g(t)\), the Ricci flow, introduced by Hamilton in (Hamilton 1982), is the PDE

\[ \frac{\partial}{\partial t} g(t) = -2\text{Ric}(g(t)). \]

This is a weakly parabolic PDE with a short-time existence theory (given initial condition \(g(0) = g_0\) where \(g_0\) is an initial smooth metric) shown by Hamilton and later by DeTurck in (DeTurck 1983).

The Ricci flow grew to prominence in G. Perelman’s use of it to resolve the Poincare conjecture. Involved in the proof were monotonicity formulas for certain energy and entropy functionals introduced by Perelman in (Perelman 2002). These monotonicity formulae by themselves are interesting, and have been used to show certain rigidity results related to the logarithmic Sobolev inequality ((Ni 2004), (Kotschwar and Ni 2009)). In Section 2 of this report we will briefly go over evolution equations for certain geometric quantities under the Ricci Flow. In Section 3 and 4, we will introduce Perelman’s energy and entropy functionals and use these evolution equations to establish the monotonicity formulae. Finally, in Section 5, we will give a survey (without proofs) of applications of these monotonicity formulae to the aforementioned rigidity results.

2 Evolution of Geometric Quantities Under Ricci Flow

Let \(M\) be an \(n\)-dimensional smooth closed manifold (in particular, it is compact and has no boundary), and let \(g(s)\) be a one-parameter family of smooth Riemannian metrics on \(M\). In this section, we will compute how certain geometric quantities evolve under the variation \(v_{ij} := \partial_s g_{ij}\). For the time being, it will be convenient to leave the variation as a general variation \(v_{ij}\), but we will later specialize this to the Ricci Flow (i.e. \(v_{ij} = \partial_t g_{ij} = -2R_{ij}\)).

2.1 Metric Inverse

Let \(g^{ij}\) denote the inverse of the matrix \(g_{ij}\). Let \(\delta^i_k\) denote the Kronecker delta, then differentiating the equation \(g^{ij}g_{jk} = \delta^i_k\), we have

\[\begin{aligned} &\partial_s\left(g^{ij}g_{jk}\right) = 0 \\ &\Rightarrow \left(\partial_s g^{ij}\right)g_{jk} + g^{ij}\left(\partial_s g_{jk}\right) = 0 \\ &\Rightarrow \left(\partial_s g^{ij}\right)g_{jk} = -g^{ij}v_{jk} \\ &\Rightarrow \partial g^{ij} = -g^{ik}g^{j\ell}v_{k\ell}. \end{aligned}\](2.1)

So in particular, if our variation is the Ricci flow, we would have that

\[ \partial_t g^{ij} = 2g^{ik}g^{j\ell}R_{k\ell}. \](2.2)

2.2 Volume Form

Let \(d\mu\) denote the Riemannian volume form induced by the metric \(g\). Then we know that

\[ d\mu = \sqrt{\det(g_{ij})}dx^1\wedge\cdots\wedge dx^n, \]

in a positively oriented local coordinate system \(\{x^i\}_{i=1}^n\).

We will use the following formula for the derivative of the determinant of a matrix which can be derived from the Liebniz formula of the determinant of a matrix. Let \(A(s)\) be a general matrix. Then we have

\[ \frac{d}{ds}\log\det{A} = \left(A^{-1}\right)^{ij}\frac{d}{ds}A_{ij}. \](2.3)

So we compute

\[\begin{aligned} \frac{d}{ds}d\mu &= \frac{d}{ds}\sqrt{\det(g_{ij})}dx^1\wedge\cdots\wedge dx^n \\ &= \frac{d}{ds}\left(\exp\left(\frac{1}{2}\log\det\left(g_{ij}\right)\right)\right) dx^1 \wedge \cdots \wedge dx^n \\ &= \frac{1}{2}\exp\left(\frac{1}{2}\log\det\left(g_{ij}\right)\right)\frac{d}{ds}\log\det\left(g_{ij}\right) dx^1 \wedge \cdots \wedge dx^n \\ &= \frac{1}{2}g^{ij}\left(\partial_s g_{ij}\right)\sqrt{\det(g_{ij})} dx^1 \wedge \cdots \wedge dx^n \\ &= \frac{1}{2}Vd\mu, \end{aligned}\](2.4)

where we used (2.3) in the penultimate line and where we are denoting \(V = g^{ij}v_{ij}\) the trace of the variation.

In particular, when the variation is the Ricci flow, we have then that

\[ \frac{1}{2}Vd\mu = \frac{1}{2} g^{ij}(-2 R_{ij})d\mu = -g^{ij}R_{ij}d\mu = -Rd\mu, \](2.5)

where \(R\) denotes the scalar curvature of the manifold with respect to the metric \(g\).

2.3 Scalar Curvature

For reasons of length, we only record the formula for the variation of the scalar curvature without showing the derivation. For the full computation, see (2.30) of chapter 2 section 5 of (Chow, Lu, and Ni 2006).

\[ \frac{\partial}{\partial s} R = -\Delta V + \text{div}(\text{div } v) - \langle v, \text{Ric}\rangle, \](2.6)

where \(v\) denotes the matrix \(v_{ij}\) and as above, \(V = g^{ij}v_{ij}\) is the trace of \(v\) with respect to the metric \(g\), \(\text{div}(\text{div } v) = \nabla_i\nabla_j v_{ij}\), and the inner product is taken with respect to \(g\) (i.e. \(\langle v, \text{Ric}\rangle = g^{ik}g^{j\ell}v_{ij}R_{k\ell}\)).

When we specialize to Ricci flow, we have that

\[\begin{aligned} \frac{\partial}{\partial t} R &= -\Delta g^{ij}(-2R_{ij}) + \nabla_i\nabla_j (-2R_{ij}) - \langle -2\text{Ric},\text{Ric}\rangle \\ &= 2\Delta R - \nabla_i 2\nabla_j R_{ij} + 2\left|\text{Ric}\right|^2 \\ &= 2\Delta R - \nabla_i \nabla_i R + 2\left|\text{Ric}\right|^2 \\ &= \Delta R + 2\left|\text{Ric}\right|^2, \end{aligned}\](2.7)

where we used the contracted second Bianchi identity \(2\nabla_jR_{ij} = \nabla_i R\) (equation (1.19) of (Chow, Lu, and Ni 2006)) in the penultimate line.

3 The Variation of the Energy Functional

Let \(M\) be a closed manifold (in particular, \(M\) is compact and has no boundary) and let \(g(s)\) be a one-parameter family of smooth Riemannian metrics on \(M\). As above, we denote \(v_{ij} = \partial_s g_{ij}\) to be the variation and \(V = g^{ij}v_{ij}\) to be the trace of the variation. Let \(f\) be a function on \(M\) that also varies with \(s\). In this section we study the variation of the functional introduced by Perelman in (Perelman 2002):

\[ \mathcal{F}(g,f) := \int_M \left(R + |\nabla f|^2\right)e^{-f}d\mu, \](3.1)

where the norm is measured with respect to the metric \(g\), i.e. \(|\nabla f|^2 = g^{ij}\nabla_i f\nabla_j f\), and the volume form \(d\mu\) is the one with respect to \(g\). Note that the variation of the total scalar curvature \(\int_M Rd\mu\) (also called the Einstein-Hilbert functional) is related to the Yamabe problem, and we can view the second term in the integrand as an Dirichlet energy-type term since

\[ \int_M |\nabla f|^2e^{-f}d\mu = 4\int_M \left|\nabla\left(e^{-f/2}\right)\right|^2d\mu. \]

One can also see that the second term is an Dirichlet energy-type term if we view \(e^{-f}d\mu\) as a weighted measure. For these reasons, \(\mathcal{F}\) is often referred to as Perelman’s energy functional.

We impose the constraint that the variation of the measure \(e^{-f}d\mu\) is zero, i.e.

\[ \frac{\partial}{\partial s}\left(e^{-f}d\mu\right) = 0. \](3.2)

We will see that we can recover the Ricci flow (up to a pullback by a diffeomorphism) as the gradient flow for \(\mathcal{F}\) coupled to a heat-type equation for \(f\) which comes from the constraint on the measure above. First, expanding out the left hand side above, and by using (2.4), we see that

\[\begin{aligned} \frac{\partial}{\partial s}\left(e^{-f}d\mu\right) &= \frac{\partial}{\partial s}e^{-f} d\mu + e^{-f}\frac{\partial}{\partial s}d\mu \\ &= -\frac{\partial}{\partial s}fe^{-f}d\mu + \frac{1}{2}e^{-f}Vd\mu \\ &= e^{-f}\left(\frac{1}{2}V - \frac{\partial}{\partial s}f\right)d\mu. \end{aligned}\]

Then, since \(e^{-f}\) is never zero, we see that the constraint (3.2) implies that \(f\) satisfies the differential equation

\[ \frac{\partial}{\partial s} f = \frac{1}{2}V. \](3.3)

Now we move on to studying the variation of \(\mathcal{F}\). We begin with

\[ \frac{\partial}{\partial s}\mathcal{F}(g,f) = \int_M \frac{\partial}{\partial s}\left(Re^{-f}d\mu\right) + \frac{\partial}{\partial s}\left(|\nabla f|^2e^{-f}d\mu\right). \](3.4)

We deal with each integrand separately. By (2.6) and the constraint (3.2), we have that

\[\begin{aligned} \int_M \frac{\partial}{\partial s}\left(Re^{-f}d\mu\right) &= \int_M \frac{\partial R}{\partial s} e^{-f}d\mu + R\frac{\partial}{\partial s}\left(e^{-f}d\mu\right) \\ &= \int_M \frac{\partial R}{\partial s}e^{-f}d\mu \\ &= -\int_M \langle v, \text{Ric}\rangle e^{-f}d\mu + \int_M \left(-\Delta V + \nabla_i\nabla_jv_{ij}\right)e^{-f}d\mu. \end{aligned}\](3.5)

Applying integration by parts twice to the last integrand above, and using that \(\nabla_j\left(e^{-f}\right) = -\nabla_jfe^{-f}\), we get that

\[\begin{aligned} \int_M \nabla_i\nabla_j v_{ij}e^{-f}d\mu &= \int_M \nabla_i\nabla_j\left(e^{-f}\right)v_{ij}d\mu \\ &= \int_M\nabla_i\left(-\nabla_jfe^{-f}\right)v_{ij}d\mu \\ &= \int_M -\nabla_i\nabla_jfe^{-f} - \nabla_jf\left(\nabla_ie^{-f}\right)v_{ij}d\mu \\ &= \int_M \left(\nabla_if \nabla_j f - \nabla_i\nabla_j f\right)e^{-f}v_{ij}d\mu. \end{aligned}\]

So (3.5) can be re-written as

\[ \int_M \frac{\partial}{\partial s}\left(Re^{-f}d\mu\right) = -\int_M \langle v,\text{Ric}\rangle e^{-f}d\mu + \int_M \left(-\Delta V + v_{ij}\nabla_i f\nabla_j f - v_{ij}\nabla_i\nabla_j f\right)e^{-f}d\mu. \](3.6)

We proceed to computing the second integral in (3.4). We have

\[\begin{aligned} \int_M \frac{\partial}{\partial s}\left(|\nabla f|^2 e^{-f}d\mu\right) &= \int_M \frac{\partial}{\partial s}\left(|\nabla f|^2\right)e^{-f}d\mu + \int_M |\nabla f|^2\frac{\partial}{\partial s}\left(e^{-f}d\mu\right) \\ &= \int_M \frac{\partial}{\partial s}\left(|\nabla f|^2\right)e^{-f}d\mu \\ &= \int_M \left(\frac{\partial}{\partial s} g^{ij}\nabla_i\nabla_j f\right)e^{-f}d\mu \\ &= \int_M \left[\left(\partial_s g^{ij}\right)\nabla_i\nabla_j f + 2g^{ij}\nabla_i f\left(\partial_s \nabla_j f\right)\right]e^{-f}d\mu \\ &= \int_M \left[-g^{ik}g^{j\ell}v_{k\ell}\nabla_i\nabla_j f + g^{ij}\nabla_i\left(\nabla_j\partial_s f\right)\right]e^{-f}d\mu \\ &= \int_M \left[-v_{ij}\nabla_if\nabla_jf + \langle\nabla f, \nabla V\rangle\right]e^{-f}d\mu, \end{aligned}\](3.7)

where we have used (2.1) and (3.3) in the penultimate and last lines above, respectively.

Note that since \(\nabla\left(e^{-f}\right) = -\nabla fe^{-f}\), the second term in the last line above can be re-written by integration by parts

\[ \int_M\langle\nabla f,\nabla V\rangle e^{-f}d\mu = \int_M -\langle\nabla\left(e^{-f}\right),\nabla V\rangle d\mu = \int_M \Delta Ve^{-f}d\mu. \]

So we get that

\[ \int_M \frac{\partial}{\partial s}\left(|\nabla f|^2e^{-f}d\mu\right) = \int_M \left(-v_{ij}\nabla_i f\nabla_j f + \Delta V\right)e^{-f}d\mu. \](3.8)

Adding the results of (3.6) and (3.8) together, see that the \(\Delta V\) and \(\nabla_i f\nabla_j f\) terms cancel out and we get that

\[ \frac{\partial}{\partial s}\mathcal{F}(g,f) = -\int_M \langle v, \text{Ric} + \nabla\nabla f\rangle e^{-f}d\mu. \](3.9)

Up to now, we haven’t specified a particular flow yet, that is, we haven’t yet specified a choice of \(v_{ij} = \partial_s g_{ij}\). This result above suggests that if we choose

\[ \frac{\partial}{\partial t}g_{ij} = v_{ij} = -2\left(R_{ij} + \nabla_i\nabla_j f\right), \](3.10)

to be the gradient flow, then we get the following monotonicity formula

\[ \frac{\partial}{\partial t}\mathcal{F}(g(t),f(t)) = 2\int_M \left|\text{Ric} + \nabla\nabla f\right|^2e^{-f}d\mu \geq 0. \](3.11)

Putting in our specific choice of \(v_{ij}\) into (3.3) we then see that \(f\) evolves as

\[ \frac{\partial}{\partial t}f = \frac{1}{2}\left(-2g^{ij}R_{ij} - 2g^{ij}\nabla_i\nabla_j f\right) = -R - \Delta f. \](3.12)

So to summarize, we have the monotonicity formula (3.11) for \(\mathcal{F}\) when the metric evolves by the gradient flow as in (3.10) under the constraint that \(e^{-f}d\mu\) is fixed, coupled with (3.12), which can be seen as a backward heat-type equation.

We can observe that (3.10) is quite similar to the Ricci flow, save for the extra Hessian term on \(f\). We can actually recover the Ricci flow via a pullback by a diffeomorphism. Note that \(\left(\mathcal{L}_{\nabla f}g\right)_{ij} = 2\nabla_i\nabla_j f\) and \(\mathcal{L}_{\nabla f}f = |\nabla f|^2\), where \(\mathcal{L}\) denotes the Lie derivative. Then we see that if \(\Psi(t): M \to M\) is a family of diffeomorphisms solving

\[ \frac{d}{dt}\Psi(t) = \nabla_{g(t)}f(t), \quad \Psi(0) = \text{id}_M, \]

then for \(g,f\) solving (3.10) and (3.12) above, then \(\bar{g}(t) = \Psi(t)^\ast g(t)\) and \(\bar{f}(t) = f \circ \Psi(t)\) satisfy the system

\[\begin{aligned} &\frac{\partial}{\partial t}\bar{g}_{ij} = -2 \bar{R}_{ij} \\ &\frac{\partial}{\partial t}\bar{f} = -\bar{\Delta}\bar{f} + |\bar{\nabla}\bar{f}|^2 - \bar{R}, \end{aligned}\](3.13)

where the barred quantities denote taking them with respect to the metric \(\bar{g}\). For reasons of length, the details of checking this computation above are left out of this report. See the proof of Lemma 5.15 of chapter 5 section 2 of (Chow et al. 2007)

One can also see that \(\mathcal{F}\) is invariant under diffeomorphisms, i.e. \(\mathcal{F}(\bar{g},\bar{f}) = \mathcal{F}(g,f)\), so we see that the same monotonicity formula (3.11) applies for \(\mathcal{F}\) under the system (3.13), which is the Ricci flow coupled with a backward heat-type equation.

We should also briefly address (again, without proofs for reasons of length) the short-time existence theory for the system (3.13) above. The Ricci flow has a well-established existence theory, which entails modifying it into a strictly parabolic equation (called the Ricci-Turck Flow) with short-time existence of solution to the corresponding initial value problem being guaranteed by the standard parabolic PDE theory (see (DeTurck 1983) for details). For the evolution equation for \(f\), setting \(u = e^{-f}\) the evolution equation becomes

\[ \frac{\partial u}{\partial \tau} = \Delta u - Ru \]

which is also a linear parabolic equation with a solution on \([0,T]\) with initial data at \(\tau = 0\) for \(u\) corresponding to final data for \(f\) at time \(T\).

4 Perelman’s Entropy Functional and Monotonicity Formula

Note that the classical entropy for a manifold (analogous to the notion of entropy from information theory), is defined by

\[ \mathcal{N} := -\int_M u\log{u}d\mu = \int_M fe^{-f}d\mu, \](4.1)

where we take \(u = e^{-f}\).

Note that under the gradient flow (3.10) and (3.12) subject to the constraint that \(\partial_t(e^{-f}d\mu) = 0\) observe that

\[ \frac{\partial}{\partial t}\mathcal{N} = \int_M \frac{\partial f}{\partial t}e^{-f}d\mu = -\int_M\left(R + \Delta f\right)d\mu = -\mathcal{F}, \]

where we use the fact that \(\int_M \Delta fe^{-f}d\mu = \int_M |\nabla f|^2e^{-f}d\mu\). We now define Perelman’s entropy which is a modification of Perelman’s energy functional that includes this classical entropy above along with a scale factor \(\tau\):

\[\begin{aligned} \mathcal{W}(g,f,\tau) &:= \int_M \left[ \tau\left(R + |\nabla f|^2\right) + f - n\right](4\pi\tau)^{-n/2}e^{-f}d\mu \\ &= (4\pi\tau)^{-n/2}\left(\tau\mathcal{F} + \mathcal{N}\right) - n\int_M (4\pi\tau)^{-n/2}e^{-f}d\mu. \end{aligned}\](4.2)

Note that the term \((4\pi\tau)^{-n/2}e^{-f}\) coincides with the heat kernel for a specific choice of \(f\) involving the Euclidean distance function.

We now consider the coupled system:

\[\begin{aligned} &\frac{\partial}{\partial t}g_{ij} = -2R_{ij} - 2\nabla_i\nabla_j f, \\ &\frac{\partial}{\partial t}f = -R - \Delta f + \frac{n}{2\tau}, \\ &\frac{\partial}{\partial t}\tau = -1. \end{aligned}\](4.3)

Then when recomputing the variation of the \(\mathcal{F}\) functional, we see that the effect of the extra \(\frac{n}{2\tau}\) term in the evolution equation for \(f\) is

\[\begin{aligned} \frac{\partial}{\partial t}\mathcal{F} &= \int_M \frac{\partial}{\partial t}(R + |\nabla f|^2)e^{-f}d\mu + \int_M (R + |\nabla f|^2)\frac{\partial}{\partial t}\left(e^{-f}d\mu\right) \\ &= 2\int_M \left|R_{ij} + \nabla_i\nabla_j f\right|^2e^{-f}d\mu + 2\int_M g^{ij}\nabla_i\left(\frac{n}{2\tau}\right)\nabla_jfe^{-f}d\mu \\ &\qquad + \int_M (R + |\nabla f|^2)\left(-\frac{\partial}{\partial t}f\right)e^{-f}d\mu + \int_M (R + |\nabla f|^2)e^{-f}(-R - \Delta f)d\mu \\ &= 2\int_M \left|R_{ij} + \nabla_i\nabla_j f\right|^2e^{-f}d\mu + \int_M (R + |\nabla f|^2)\left(R + \Delta f - \frac{n}{2\tau}\right)e^{-f}d\mu \\ &\qquad + \int_M (R + |\nabla f|^2)(-R - \Delta f)e^{-f}d\mu \\ &= 2\int_M \left|R_{ij} + \nabla_i\nabla_j f\right|^2e^{-f}d\mu - \frac{n}{2\tau}\mathcal{F}. \end{aligned}\](4.4)

We also have that the variation of the classical entropy under the above system,

\[\begin{aligned} \frac{\partial}{\partial t}\mathcal{N} &= \int_M \left(\frac{\partial}{\partial t}f\right)e^{-f}d\mu + \int_M f\frac{\partial}{\partial t}\left(e^{-f}d\mu\right) \\ &= \int_M \left(-R - \Delta f + \frac{n}{2\tau}\right)e^{-f}d\mu + \int_M f\left(-\frac{\partial}{\partial t}f\right)e^{-f}d\mu \\ &\qquad + \int_M fe^{-f}(-R -\Delta f)d\mu \\ &= -\int_M (R + \Delta f)e^{-f}d\mu + \frac{n}{2\tau}\int_M e^{-f}d\mu \\ &\qquad + \int_M f\left(R + \Delta f - \frac{n}{2\tau}\right)e^{-f}d\mu + \int_M fe^{-f}(- R - \Delta f)d\mu \\ &= -\int_M (R + |\nabla f|^2)e^{-f}d\mu + \frac{n}{2\tau}\int_M e^{-f}d\mu - \frac{n}{2\tau}\int_M fe^{-f}d\mu \\ &= -\mathcal{F} + \frac{n}{2\tau}\int_M e^{-f}d\mu - \frac{n}{2\tau}\mathcal{N}. \end{aligned}\](4.5)

Then using (4.4) and (4.5), and imposing the constraint that \(\int_M(4\pi\tau)^{-n/2}e^{-f}d\mu = 1\), we compute

\[\begin{aligned} \frac{\partial}{\partial t}\mathcal{W} &= \frac{\partial}{\partial t}\left[(4\pi\tau)^{-n/2}\left(\tau\mathcal{F} + \mathcal{N}\right)\right] + \frac{\partial}{\partial t}n\int_M(4\pi\tau)^{-n/2}e^{-f}d\mu \\ &= \left(\frac{\partial}{\partial t}(4\pi\tau)^{-n/2}\right)\left(\tau\mathcal{F} + \mathcal{N}\right) + (4\pi\tau)^{-n/2}\frac{\partial}{\partial t}\left(\tau\mathcal{F} + \mathcal{N}\right) \\ &= (4\pi\tau)^{-n/2}\frac{n}{2\tau}\left(\tau\mathcal{F} + \mathcal{N}\right) + (4\pi\tau)^{-n/2}\left(-\mathcal{F} + \tau\frac{\partial}{\partial t}\mathcal{F} + \frac{\partial}{\partial t}\mathcal{N}\right) \\ &= (4\pi\tau)^{-n/2}\left(\frac{n}{2\tau}\left(\tau\mathcal{F} + \mathcal{N}\right) + 2\tau\int_M\left|R_{ij} + \nabla_i\nabla_j f\right|^2e^{-f}d\mu - 2\mathcal{F}\right.\\ &\left.\qquad\qquad\qquad\qquad- \tau\frac{n}{2\tau}\mathcal{F} - \frac{n}{2\tau}\mathcal{N} + \frac{n}{2\tau}\int_Me^{-f}d\mu\right) \\ &= (4\pi\tau)^{-n/2}\left(2\tau\int_M\left|R_{ij} + \nabla_i\nabla_j f\right|^2e^{-f}d\mu - 2\mathcal{F} + \frac{n}{2\tau}\int_M e^{-f}d\mu\right). \end{aligned}\]

Then completing the square we discover the monotonicity formula for Perelman’s entropy functional,

\[ \frac{\partial}{\partial t}\mathcal{W} = 2\tau\int_M \left|R_{ij} + \nabla_i\nabla_j f - \frac{1}{2\tau}g_{ij}\right|^2e^{-f}d\mu \geq 0, \](4.6)

for \(\tau > 0\).

5 Applications to Rigidity Results

Though it is out of scope to go into the details for this report, we mention that Perelman used the above monotonicity formula for the entropy to prove what is called the no-local collapsing theorem, which was used to understand the singularity models of Ricci flow in the proof of the Geometrization/Poincare conjecture.

In the final part of this section and report, we survey a suite of similar monotonicity formulae to static manifolds (that is, manifolds not evolving by Ricci flow) by Ni and collaborators and their applications to rigidity problems in geometry.

The first result is by Ni in (Ni 2004):

That is, the entropy \(\mathcal{W}\) is monotone decreasing along the heat equation. Note that change in sign from that of Perelman’s result and the absence of the scalar curvature term in this entropy defined for the static case. Ni also extended the above monotonicity result to the class of complete noncompact manifolds with nonnegative Ricci curvature.

The logarithmic Sobolev inequality (which is a dimensionless version of the usual Sobolev inequality) can be written as

\[ \int \varphi^2\log\varphi d\mu \leq \int |\nabla\varphi|^2d\mu, \](5.4)

for \(\varphi\) such that \(\int \varphi^2 d\mu = 1\).

Part of Perelman’s work with the entropy functional is to show that \(\mathcal{W} \geq 0\) at some time \(\tau_0\) is equivalent to the logarithmic Sobolev inequality above being valid on the manifold \(M\). With that fact, one can then use Ni’s monotonicity formula above to show that \(\mathcal{W} \equiv 0\) after that time, which Ni uses to establish the following rigidity result for the geometry of the manifold:

Later, Kotschwar and Ni in (Kotschwar and Ni 2009) generalized the above result to a certain \(p\)-heat equation for \(p > 1\), where the Laplacian operator in (5.1) above is replaced by the \(p\)-Laplacian given by

\[ \Delta_p u := \text{div}(|\nabla u|^{p-2}\nabla u), \]

and we consider the following \(p\)-heat equation

\[ \frac{\partial}{\partial t}u^{p-1} = (p-1)^{p-1}\Delta_pu. \](5.5)

We define \(f\) analogously as before following the form of the fundamental solution to the above equation as follows:

\[ u^{p-1} = \frac{1}{\pi^{n/2}(p^{\ast p-1}p)^{n/p}}\frac{\Gamma(n/2+1}{\Gamma(n/p^\ast + 1}e^{-f}t^{-n/p}. \](5.6)

Then choosing \(f\) such that \(\int_M u^{p-1}d\mu = 1\), we can define the following \(p\)-entropy:

\[ \mathcal{W}_p(g,f,t) := \int_M \left(t|\nabla f|^p + f - n \right)u^{p-1}d\mu. \](5.7)

Then Kotscwhar and Ni find the following monotonicity formula for this entropy.

Note that the tensor \(A\) is obtained via linearizing equation (5.5) above and is given in local coordinates by \(A^{ij} = g^{ij} + \frac{p-2}{|\nabla f|^2}f^if^j\). Then taking the norm of a 2-tensor with respect to \(A\) is given by \(|T|_A^2 = A^{ik}A^{j\ell}T_{ij}T_{k\ell}\).

Analogously we also have the \(p\)-logarithmic Sobolev inequality,

\[ \int |\varphi|^p \log{|\varphi|^p} \leq \frac{n}{p}\log\left(C_{p,n}\int |\nabla\varphi|^p\right), \](5.9)

for \(p > 1\), a constant \(C_{p,n}\) depending on \(p\) and \(n\) and \(\varphi\) such that \(\int|\varphi|^p = 1\).

Then similarly one can show that \(\mathcal{W}_p \geq 0\) is equivalent to the \(p\)-logarithmic Sobolev inequality being valid on the manifold (Proposition 6.18 of (Kotschwar and Ni 2009)). Then finally, Kotschwar and Ni establish the following rigidity result:

References