Questions tagged [geometric-analysis]
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9 questions from the last 365 days
6
votes
2
answers
390
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Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
0
votes
0
answers
65
views
Regularity of Metric when defining C^k norms
Given $(M^n, g)$ closed riemannian manifold, I am wondering about the definition of the $C^k$ norms with respect to the metric, and how these norms depend on $g$. For example, I would assume that if $...
- 337
2
votes
0
answers
72
views
Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
- 337
14
votes
1
answer
572
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Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
0
votes
0
answers
40
views
Is the principal eigenvalue of the clamped plate minimized by the equilateral triangle among all triangles of a given area?
Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$
(bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by
$$\Delta^2 u = \lambda u $$
$$ u|_{\partial \...
- 3,245
3
votes
0
answers
67
views
How powerful are sequences of Steiner symmetrizations?
I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
- 171
1
vote
0
answers
56
views
Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
- 383
5
votes
0
answers
337
views
Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions
Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
- 319
3
votes
1
answer
283
views
A compact Riemann surface with a finite set of points removed is parabolic
A Riemann surface $\mathcal{R}$ is called parabolic if it is not compact and doesn't carry a negative non-constant subharmonic function, and is called hyperbolic if it carries a negative non-constant ...
- 438