Questions tagged [geometric-analysis]
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166 questions
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Green's function for *GJMS* operator
Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
- 327
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A priori $C^0$ estimates for a semi-linear vector Poisson equation
Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0
$$
Here $a_i:...
user34072
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In a short time $0<t<T$, does the $C^{1,\alpha}$ norm of mean curvature flow depend only on $T$ and the $C^{1,\alpha}$ norm of initial surface?
Given a smooth compactly embedded initial hypersurface $M_0$ and let $M_t$ ($0<t<T$) be a smooth solution of mean curvature flow starting from $M_0$. Then how does the regularity of $M_t$ depend ...
- 1,350
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1
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Hausdorff measure and projections
Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. ...
- 520
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2
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Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature
A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...
- 5,984
3
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1
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Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?
Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...
- 131
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434
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Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
- 601
3
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2
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Estimates on a heat process with fixed boundary data and zero initial conditions
Consider the following heat process:
For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that
$$
\partial_t p(...
- 337
3
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1
answer
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Different forms of Bonnesen's strong isoperimetric inequality in the plane.
I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve:
$(R_{out} - R_{in})^2 \leq \pi^2 (L^2 - 4\pi A),$
where $R_{out}$ ...
- 2,641
2
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2
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Is there a combinatorial analogue of the Kazdan Warner theorem?
First let me state a result of Kazdan and Warner
Let $M$ be a compact orientable two dimensional manifold.
Let $f:M \rightarrow \mathbb{R}$ be a function that has the same
sign as the Euler ...
- 3,245
3
votes
0
answers
242
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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 $\chi(M)$,...
- 3,245
4
votes
1
answer
523
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Geometric bound on the first eigenvalue of Laplace-Beltrami on forms
Let $M$ be a closed Riemannian manifold and $\Delta$ its Laplace-Beltrami operator. Then we have Yau's estimate on the first (non-zero) eigenvalue $\lambda_1>0$ of $\Delta$ acting on functions in ...
- 41
1
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435
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What is the shape of a tight open trefoil?
Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically.
The problem is ...
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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 ...
2
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Diameters of the images of two balls under a function
Let $ \Omega $ be an open and bounded subset of $ \mathbb{R^n} $, and let $ f:\Omega \to \mathbb{R} $ be a continuous function. I'm looking for some (preferably, minimal) conditions on $ f $ under ...
- 520
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Finding the lift of a curve under some assumptions
Let $f:\Omega\subseteq\mathbb{R}^n\to\mathbb{R}^n$ be a Lipschitz function and $h$ be a vector in $\Omega$. Assume that $0\in\Omega$ and $f(0) = 0$. Also, let $\sigma:[0,1]\to\mathbb{R}^n$ be the ...
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