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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 $\...
John McManus's user avatar
2 votes
0 answers
664 views

Reference request - Texts on geometric analysis with exercises

I’ve recently been studying some Riemannian geometry and geometric analysis, however I have found it difficult to find resources with exercises to practice. It seems that many textbooks past the ...
4 votes
0 answers
114 views

How can I numerically solve the Laplace equation with cohomological data?

Consider the problem of solving for $u$ where $-\Delta u = f$, $[u] = [g]$ where $[\cdot]$ denotes cohomology class and $u, f, g$ are $p$-forms on a Riemannian manifold $M$. If $g$ instead was ...
Aidan Backus's user avatar
2 votes
0 answers
127 views

Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric ...
Laithy's user avatar
  • 969
2 votes
0 answers
141 views

For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $...
Laithy's user avatar
  • 969
2 votes
0 answers
269 views

Solvability of a PDE involving the Dirichlet-to-Neumann operator

Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric). Let $N: L^2(\...
Laithy's user avatar
  • 969
13 votes
2 answers
2k views

Is there a solution of the Yamabe problem using Ricci flow?

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
Hollis Williams's user avatar
3 votes
2 answers
437 views

Proof of Isoperimetric Inequality using Curve Shortening Flow

I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...
Hollis Williams's user avatar
4 votes
0 answers
97 views

One-dimensional harmonic map flow with low regularity

My question is the following: What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in ...
Matthias Ludewig's user avatar
3 votes
2 answers
450 views

Resources on Elliptic Boundary Value Problems on manifolds

My situation: I am currently trying to understand Uhlenbecks results on the Yang Mills equation. One of the most common notions in this paper is that of an elliptic system or an elliptic boundary ...
Peter Wildemann's user avatar
11 votes
2 answers
1k views

Non-compact manifolds of positive/non-negative Ricci curvature

Consider a non-compact complete Riemannian manifold $(M, g)$ with smooth compact boundary $\partial M$. Suppose also that $M \setminus \partial M$ has positive/non-negative Ricci curvature. My ...
user111682's user avatar
1 vote
0 answers
55 views

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 ...
student's user avatar
  • 1,350
2 votes
1 answer
95 views

literature/reference request for estimates of first eigenvalue of certain Schrodinger operator on compact surfaces

On compact Riemannian surfaces (say without boundary), the Schrodinger operator I am interested in is of the form $-\Delta+2\kappa$, where $\kappa$ is the Gauss curvature. For minimal surfaces in $\...
Piojo's user avatar
  • 783
5 votes
2 answers
1k views

Self-contained book on Ricci Flow/Geometric Analysis

Can someone please tell me whether there is any self-contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self-contained I mean it does not assume that ...
Bingo's user avatar
  • 789
4 votes
3 answers
2k views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
Axiom's user avatar
  • 520
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
6 votes
2 answers
554 views

Hamilton-Ivey pinching in dimension 4

I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. ...
Brian Klatt'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
7 votes
1 answer
554 views

Lower bound on $L^2$ norm of mean curvature in general dimensions

Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$ $$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and that this is saturated ...
Rbega's user avatar
  • 2,299
22 votes
2 answers
5k views

Surface equivalent of catenary curve

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the shape taken by an ...
Joseph O'Rourke's user avatar
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