Heat kernel estimates on manifolds with ends with mixed boundary condition

TL;DR

The paper addresses the problem of obtaining sharp two-sided heat kernel bounds on weighted manifolds with finitely many ends under mixed boundary conditions. It introduces a harmonic profile $h$ on an open set $\Omega$ with boundary, and uses an $h$-transform to reduce the problem to a connected-sum of Harnack ends, enabling the application of GS5-type heat-kernel techniques in the transformed space. By relating the global kernel $p(t,x,y)$ to the transformed kernel via $p(t,x,y)=h(x)h(y)p_{\Omega,h^2}(t,x,y)$, and carefully analyzing end Green functions, volumes, and Harnack structure, the authors derive explicit two-sided bounds valid for all $t>0$ and $x,y\in\Omega$, with clear dependence on end geometry and boundary conditions. The work also provides detailed examples with cones and paraboloid-like ends to illustrate how Dirichlet versus Neumann sides and end apertures shape the long-time decay of the heat kernel, thereby unifying local boundary behavior with global end-geometry. This framework advances the understanding of mixed-boundary heat kernels on manifolds with ends and has applications to diffusion processes in domains with complex geometries.

Abstract

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and Saloff-Coste by allowing for Dirichlet boundary condition. The proof requires the construction of a global harmonic function which is then used in the $h$-transform technique.

Figures (9)

• Figure 1: A solid subset of $\mathbb R^3$: a complete manifold with boundary.
• Figure 2: Sketch of a planar, unbounded, complete manifold $M$ (light red) with boundary $\delta M$ (blue and dark red) with three conic ends. Dirichlet boundary is depicted in blue, Neumann boundary in red. Corners should be rounded so that $M$ is really a (smooth) manifold with boundary, though actually it does not matter; see Appendix \ref{['corners']}.
• Figure 3: Same $M$ as in Figure \ref{['fig:cones1']}, but with different boundary conditions.
• Figure 4: An $M$ with an end that is a cone of aperture zero.
• Figure 5: Sketch (corners should be rounded) of the complete manifold $M$ (dark red) and its submanifold $\Omega$ (light red and red boundary) with "Dirichlet boundary" $\partial \Omega\subseteq \delta M$, not part of $\Omega$, highlighted in blue.

Theorems & Definitions (57)

• Definition 2.1: The Sobolev space $W^1_0(V)$
• Definition 2.2: Heat equation on $M$
• Theorem 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Lemma 3.5
• proof
• proof : Proof of Theorem \ref{['thm-profile']}.