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9 votes
2 answers
655 views

Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
Pablo Lessa's user avatar
  • 4,304
1 vote
1 answer
525 views

Hamilton's Proof of the Tensor Maximum Principle

My questions come from Richard Hamilton's Three-Manifolds with Positive Ricci Curvature paper. I'm trying to work through parts of the paper so I can better understand the Ricci Flow for my research. ...
Brian Klatt's user avatar
2 votes
0 answers
102 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
A curious cat's user avatar
4 votes
1 answer
699 views

Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
user avatar
3 votes
0 answers
242 views

What is known about analogous results of Kazdan and Warner in higher dimensions?

First let me state a Theorem due to Kazdan and Warner: ``Let M be a compact two dimensional orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the same sign as $\chi(M)$,...
Ritwik's user avatar
  • 3,245
7 votes
4 answers
3k views

How does curvature change under perturbations of a Riemannian metric?

Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
Tom LaGatta's user avatar
  • 8,512

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