Related papers: Small time heat kernel asymptotics at the sub-Riem…

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict…

Let $\Omega$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $\Omega$ has infinite measure,…

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…

We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green's functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike…

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(\epsilon,\delta)$-domains $\Omega\subset\mathbb{R}^n$ with a $d$-set…

We consider Schroedinger operators on compact and non-compact (finite) metric graphs. For such operators we analyse their spectra, prove that their resolvents can be represented as integral operators and introduce trace-class…

We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic…

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale…

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for…

We build a systematic calculational method for the covariant expansion of the two-point heat kernel $\hat K(\tau|x,x')$ for generic minimal and non-minimal differential operators of any order. This is the expansion in powers of dimensional…

We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider subsurfaces of a three dimensional contact sub-Riemannian manifolds, and using a family of taming Riemannian metric, we obtain a pure sub-Riemannian…

We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local…

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed…

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian…

The paper deals with minimum energy problems in the presence of external fields on a locally compact space $X$ with respect to a function kernel $\kappa$ satisfying the energy and consistency principles. For quite a general (not necessarily…

For an embedded submanifold $\Sigma\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we…

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in…

Ambrose and Singer characterized connected, simply-connected and complete homogeneous Riemannian manifolds as Riemannian manifolds admitting a metric connection such that its curvature and torsion are parallel. The aim of this paper is to…

Let $G$ be a compact connected Lie group equipped with a bi-invariant metric. We calculate the asymptotic expansion of the heat kernel of the laplacian on $G$ and the heat trace using Lie algebra methods. The Duflo isomorphism plays a key…