[Paper Review] On the Heat Kernel under the Ricci Flow Coupled with the Harmonic Map Flow

Mihai Băileșteanu|arXiv (Cornell University)|Aug 31, 2013

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper establishes a bound on the heat kernel under the Ricci-harmonic flow on closed Riemannian manifolds of dimension at least 3, using Sobolev embedding constants and time-dependent metric evolution. When the initial scalar curvature satisfies $ R > \alpha(0)|\nabla\phi(0)|^2 $, the bound recovers the classical Gaussian-type decay seen in the fixed metric case, improving prior results by using sharper constants.

ABSTRACT

We estimate the heat kernel on a closed Riemannian manifold $M$, with $dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result depends on some constants arising from a Sobolev imbedding theorem. In a special case, when the scalar curvature satisfies a certain natural inequality, we obtain, as a corollary, a bound similar to the one known for the fixed metric case.

Motivation & Objective

• To derive a pointwise upper bound for the heat kernel under the Ricci-harmonic flow, a coupled system of Ricci flow and harmonic map flow.
• To analyze how the time-dependent geometry, driven by both metric evolution and map evolution, affects heat kernel behavior.
• To identify conditions under which the heat kernel exhibits decay similar to the static metric case.
• To improve upon existing heat kernel bounds under Ricci flow by incorporating coupling terms via Sobolev embedding constants.

Proposed method

• Derives a differential inequality for the $ L^2 $-norm of the heat kernel using the heat equation and properties of the metric and map evolution.
• Applies a Sobolev imbedding inequality with best constants $ A(t) $ and $ B(t) $, which depend on the geometry and curvature bounds.
• Uses an integrating factor method with a time-dependent function $ H(\tau) $ derived from the ratio $ B(\tau)/A(\tau) $ and a correction term involving $ m_0 $ and $ c_n $.
• Establishes separate bounds on the heat kernel over two time intervals: $ [s, (s+t)/2] $ and $ [(s+t)/2, t] $, then combines them via multiplication.
• Imposes the condition that $ S = R - \alpha|\nabla\phi|^2 $ is bounded below at $ t=0 $, ensuring $ S > 0 $ is preserved, which simplifies the bound.
• Applies Hölder’s inequality and the semigroup property of the heat kernel to control the $ L^2 $-norms and derive the final estimate.

Experimental results

Research questions

• RQ1 Under what conditions does the heat kernel under the Ricci-harmonic flow exhibit decay similar to the fixed metric case?
• RQ2 How do the best constants in the Sobolev embedding theorem influence the heat kernel bound under time-dependent metrics?
• RQ3 Can the heat kernel bound be improved by incorporating coupling terms from the harmonic map flow?
• RQ4 What role does the quantity $ S = R - \alpha|\nabla\phi|^2 $ play in preserving geometric control during the flow?
• RQ5 Is it possible to derive Gaussian-type bounds under more general curvature or coupling conditions?

Key findings

• The heat kernel satisfies the bound $ G(x,t;y,s) \leq \frac{C_n}{\left(\int_s^{(s+t)/2} \left(\frac{m_0 - c_n\tau}{m_0}\right)^{-2} \frac{e^{\frac{2}{n}H(\tau)}}{A(\tau)} d\tau\right)^{n/4} \left(\int_{(s+t)/2}^t \frac{e^{-\frac{2}{n}H(\tau)}}{A(\tau)} d\tau\right)^{n/4}} $, where $ H(\tau) $ is an antiderivative of $ \frac{B(\tau)}{A(\tau)} - \frac{3}{4}(m_0 - c_n\tau)^{-1} $.
• When $ R > \alpha(0)|\nabla\phi(0)|^2 $, the bound reduces to $ G(x,t;y,s) \leq \frac{\tilde{C}_n}{(t-s)^{n/2}} $, matching the classical Gaussian decay rate.
• The constant $ \tilde{C}_n = \left(\frac{4K(n,2)}{n}\right)^{n/2} $, where $ K(n,2) $ is the best constant in the Sobolev embedding $ W^{1,2} \hookrightarrow L^p $ for $ p = \frac{2n}{n-2} $.
• The bound is derived without curvature assumptions, relying only on the Sobolev constants and the non-increasing nature of $ \alpha(t) $.
• The result improves upon earlier bounds under Ricci flow by using sharper constants derived from the coupled flow structure.
• The proof technique extends the method of Q. Zhang and Perelman via integrating factors and energy estimates, adapted to the time-dependent, coupled setting.