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1 vote
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87 views

Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request

Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian"). Where can I find proved ...
Alex M.'s user avatar
  • 5,407
2 votes
0 answers
113 views

Is this $1$-form harmonic?

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
Eduardo Longa's user avatar
0 votes
2 answers
289 views

Derivations of $\chi^{\infty}(M)$ which are elliptic operator

What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
Ali Taghavi's user avatar
3 votes
1 answer
177 views

Does the space of harmonic forms change continuously with the metric?

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...
Asaf Shachar's user avatar
  • 6,741
5 votes
1 answer
132 views

Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds

In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
Alex M.'s user avatar
  • 5,407
17 votes
4 answers
3k views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
mdg's user avatar
  • 376