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Let $\Delta$ be the negative-definite Laplacian. Let $f \in V$ and $x \in M$ be arbitrary, but fixed. Is it true that ${\rm Re} \ (\Delta f) (x) \overline {f (x)} \le 0$? … In its support, I have two arguments that lend it some plausibility: 1) if $f \in V$ is an eigenfunction of the Laplacian, the conjecture is true. …

I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). … In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes the angular coorinates taken together), the Laplacian looks like $\Delta = \frac {\partial ^2} {\ …

(Let me add that the heat equation differs by a factor of $\frac 1 2$ in front of the Laplacian between the two references above, but the versions of Varadhan's formula are identical, which makes me believe …

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold. … point me to texts doing the same kind of construction (not necessarily using densities) in vector bundles over arbitrary (i.e. not necessarily compact anymore) Riemannian manifolds for the connection Laplacian …

If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs extension, that we shall also denote $L$. …

Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively …

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial u}{\par …

If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat o …

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the Laplacian … (In particular, if the whole expression inside the Laplacian were $O(r ^{2 + \varepsilon})$ for some $\varepsilon > 0$, the answer would be affirmative.) …

If $M$ is a complete Riemannian manifold and $L$ is the Friedrichs extension of the Laplacian $-\Delta$, then it is known (first proven by Gaffney in the '50) that $C_0 ^\infty (M)$ is a core for $L$. …

One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the …

Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p …