Related papers: Heat content and horizontal mean curvature on the …

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are…

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give…

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat…

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique…

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body $K_0 \subseteq \mathbb{R}^{2n}$, for $t$-graphs on a bounded domain $\Omega$ in the Heisenberg group $\mathbb{H}^n$. When the prescribed datum…

Rapidly rotating Rayleigh-B\'enard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk…

We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian…

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators $\mathscr L^s$ and…

Using the quantum Hamiltonian for a gravitational system with boundary, we find the partition function and derive the resulting thermodynamics. The Hamiltonian is the boundary term required by functional differentiability of the action for…

In a previous work, the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower…

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any…

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a…

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain…

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given…

We study constant mean curvature surfaces in the three-dimensional Heisenberg group. We prove that a constant mean curvature surface in a neighborhood of non-umbilic point is described by some solution of a sinh-Gordon equation subject to a…

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic…

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$…

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also…

In this paper, we prove first-order asymptotics on a bounded open set of the heat content when the ambient space is an ${\sf RCD}(K,N)$ space, under a regularity condition for the boundary that we call measured interior geodesic condition…

In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this…