All Questions
Tagged with geometric-analysis reference-request
21 questions
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 ...
- 151k
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 ...
- 5,146
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 ...
- 111
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 ...
- 4,304
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 ...
- 8,512
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 ...
- 2,299
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. ...
- 787
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 ...
- 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 ...
- 520
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 $0$ ...
user14166
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 ...
- 708
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 ...
- 9,853
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 ...
- 5,146
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 ...
- 381
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 $\...
- 783
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 ...
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 ...
- 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 $g_0$, $...
- 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(\...
- 969
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
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 ...
- 1,350