The Atiyah-Patodi-Singer theorem for perturbed Dirac operators on even-dimensional manifolds with bounded geometry.
The Bochner Laplacian, Riemannian submersions, heat content asymptotics, and heat equation asymptotics
The evolution of the Dirac field in cuved space-times.
The heat equation and harmonic maps of complete manifolds.
The heat equation on manifolds as a gradient flow in the Wasserstein space
We study the gradient flow for the relative entropy functional on probability measures over a riemannian manifold. To this aim we present a notion of a riemannian structure on the Wasserstein space. If the Ricci curvature is bounded below we establish existence and contractivity of the gradient flow using a discrete approximation scheme. Furthermore we show that its trajectories coincide with solutions to the heat equation.
The heat semigroup acting on tensors or differential forms with values in vector bundle
The logarithmic gradient of the kernel of the heat equation with drift on a Riemannian manifold.
The Logarithmic Hardy-Littlewood-Sobolev Inequlity and Extremals of Zeta Functions on Sn.
The resolvent expansion for the signature operator on a manifold with a conic singular stratum
The signature theorem for manifolds with metric horns
Two examples of fattening for the curvature flow with a driving force
We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.