This paper describes results characterizing the range of the time-t heat operator on various manifolds, including Euclidean spaces, spheres, and hyperbolic spaces. The guiding principle behind these results is this: The functions in the range of the heat operator should be, roughly, those functions having an analytic continuation to the appropriate complexified manifold with growth in the imaginary directions at most like that of the time-t heat heat kernel.
In this paper, we compute the first two equivariant heat kernel coefficients of the Bochner Laplacian on differential forms. The first two equivariant heat kernel coefficients of the Bochner Laplacian with torsion are also given. We also study the equivariant heat kernel coefficients of nonmininmal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.
We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace--Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise…
Let $G$ be a connected compact Lie group. We study the heat operator of a $G$-transversally elliptic operator. After we review the spectral properties of a $G$-transversally elliptic operator, we define the character, that is a distribution on $G$ generalizing the trace of the heat operator to the $G$-equivariant case. The main theorem of this paper gives the estimate of $f_α(t)$, which essentially determines the convergence of the character.
We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffr…
We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part $-\N^μ\N_μ$. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system…
We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat invariants associated with these operators. As another application, we present new explicit formulas for the matrix Korteweg-de Vries hierarchy.
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the drift field and the curvature-like invariants of the optimal control problem associated with the diffusion operator.
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside…
The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2. A characterization of p-hyperbolicity, which is of independent interest, is also proved.
We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. This yields short time existence for a class of semi-linear equations of fourth order, including for example the Cahn-Hilliard equation. We also obtain asymptotics of the solutions near the edge singularity.
We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hörmander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order …
We study the heat trace for both the drifting Laplacian as well as Schrödinger operators on compact Riemannian manifolds. In the case of a finite regularity potential or weight function, we prove the existence of a partial (six term) asymptotic expansion of the heat trace for small times as well as a suitable remainder estimate. We also demonstrate that the more precise asymptotic behavior of the remainder is determined by and conversely distinguishes higher (Sobolev) regularity on the potential or weight function. In the case of a smooth weight function, we determine the full asymptotic expan…
We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coeff…
In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact $S^7$ case, we give a…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particular to CR Sasakian manifolds with Tanaka-Webster-Ricci curvature bounded from below and Carnot groups of step two.
Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formulæfor several geometric and spectral invariants associated to a Dirac-type operator ð_X on X are presented. Considered in detail are: the index of ð_X, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as…
In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe a joint result with Xianzhe Dai on an index theorem for Toeplitz operators on odd dimensional manifolds with boundary.
The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universal heat kernel expansion for formally selfadjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion imp…
We study the geometry associated to the Grusin operator G=Δ_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory distance d_{CC}, associated to this operator. The main result in the second part is to give Gaussian bounds for the heat kernel K_{t} in terms of the Carnot-Caratheodory distance. In particular we obtain the following estimate |k_{t}(ζ,η)|\leq C t^{-\frac{n}{2}-1}\min(1+\frac{d_{CC}(…
I prove the existence of small time heat expansion for the Laplace operator on an analytic hypersurface with an isolated singularity. First we obtain a local parametrization of the hypersurface near the singularity. We introduce the notion of quasihomogeneous tangent cone. Then perturb the parametrization of the cone employing a Newton scheme and obtain a parametrization with functions of specific form. These allow us to obtain local models for the Laplace operator near the singularity. These are operators with irregular singularities. We derive the estimates required by singular asymptotics.
We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schrödinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operat…
