On monotonicity of heat kernels: a new example and counterexamples

TL;DR

This work addresses whether the heat kernel $K_t$ on a Riemannian manifold decreases along geodesics from a fixed point for all times. It develops a framework based on spectral expansions and the parabolic maximum principle to analyze geodesic monotonicity, yielding a new non-radial example—the honeycomb flat torus—where $K_t$ is geodesically decreasing for every $t>0$. It also proves a sharp classification: among flat tori, monotonicity occurs only for rectangular or regular triangular lattices, while flat Klein bottles fail at large times; for generic metrics, monotonicity generally fails as $t oty$. The paper further derives necessary spectral conditions tied to eigenvalue multiplicities and explores the double-eigenvalue case, showing strong rigidity that constrains the manifold's geometry (e.g., leading to product structures) and clarifying when monotonicity can occur.

Abstract

We discover a new, non-radial example of a manifold whose heat kernel decreases monotonically along all minimal geodesics. We also classify the flat tori with this monotonicity property. Furthermore, we show that for a generic metric on any smooth manifold the monotonicity property fails at large times. This answers a recent question of Alonso-Orán, Chamizo, Martínez, and Mas.

Figures (4)

• Figure 1: Two fundamental domains for the standard honeycomb lattice. The parallelogram is spanned by the basis vectors $e_1=(1,0)$ and $\zeta=(-\frac{1}{2},\frac{1}{2}\sqrt{3})$. The boundary of the regular hexagon $C$ corresponds to the cut locus of the origin in the honeycomb torus ${\mathbb R}^2/\Lambda$. The lattice is symmetric under reflection across the three lines that form the sides of $D$.
• Figure 2: Fundamental parallelogram of a lattice $\Lambda$ without special symmetries. The lattice basis is $\{(1,0),(-a,b)\}$, where $0<a\le\frac{1}{2}$, $b>0$, and $a^2+b^2>1$. The thin lines correspond to the cut locus of the origin in the torus ${\mathbb R}^2/\Lambda$. Geodesic monotonicity fails on the segment of the vertical geodesic emanating from $(0,0)$ that lies inside the shaded triangle.
• Figure 3: Fundamental parallelogram of an isosceles lattice $\Lambda$ with basis $\{(1,0),(-a,b)\}$, where $0<a<\frac{1}{2}$, $b>0$, and $a^2+b^2=1$. In addition to translation along the basis vectors, the lattice is symmetric under reflection at the diagonals (dashed), both of which are longer than the sides. The thin lines inside the parallelogram correspond to the cut locus of the origin in ${\mathbb R}^2/\Lambda$. The point $z^*$ is equidistant to $O$, $A$, and $B$. Geodesic monotonicity fails on the segment of the geodesic connecting $O$ to $z^*$ that lies inside the small shaded triangle.
• Figure 4: Fundamental rectangle for a flat Klein bottle of height $b>1$. The left and right edges are identified to form a cylinder, and the top and bottom circles are glued by the orientation-reversing isometry $\psi(x)=-x$. Given $\xi\in (0,\frac{1}{2})$, the thin lines correspond to the cut locus of $(\xi,0)$ in the Klein bottle. Geodesic monotonicity fails on the segment of the vertical geodesic emanating from $(\xi,0)$ that lies inside the shaded triangle.

Theorems & Definitions (36)

• Theorem 1.1: Radially decreasing heat kernels CYChAnderssonNSSACMM
• Theorem 1.2: Symmetric decrease about a point ACMM
• Definition : Geodesic monotonicity
• Theorem 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop-honeycomb']}