All Questions
24 questions
3
votes
0
answers
158
views
Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
- 969
5
votes
1
answer
310
views
Lee-Parker Yamabe problem proposition 4.6
I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
- 457
7
votes
1
answer
506
views
Existence and estimates of Green's function on Riemannian manifold
In Yau and Schoen's differential geometry,in Ch5 before Thm 3.5,the author says
When $R$(scalar curvature of a manifold M)$>0$,there exists a unique Green's function $G$ to the operator $L=-\Delta+...
- 217
7
votes
2
answers
365
views
What's the limit of a sequence of harmonic maps between manifolds?
Let ${M}, \, {N}$ be two Riemannian manifolds, and let $u_n: {M} \to {N}$ be a sequence of harmonic maps.
Question. Suppose that $u_n$ converges uniformly to a (necessarily continuous) function $u$. ...
- 438
6
votes
1
answer
294
views
Solving $\Delta \text{tr}(h) - \mathrm{div}(\mathrm{div}(h)) + \text{tr}(h) = f$ on $S^2$
$\DeclareMathOperator\ddiv{div}\DeclareMathOperator\tr{tr}\newcommand{\conf}{\mathrm{conf}}$Consider this PDE on a symmetric tensor $h$ on $S^2$:
$$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$
...
- 969
7
votes
1
answer
281
views
Existence of harmonic maps onto the $n$-sphere
Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g_{st})$ be the standard round $n$-sphere.
Whether there exists a non-...
- 1,799
2
votes
0
answers
269
views
Solvability of a PDE involving the Dirichlet-to-Neumann operator
Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric).
Let $N: L^2(\...
- 969
7
votes
1
answer
490
views
Yau's conjecture on nodal sets for manifolds with boundary
I've just read a review paper about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds.
Briefly, if $\phi_\lambda$, $\lambda$ are an eigenpair for the Laplace-...
- 840
7
votes
0
answers
187
views
Level Sets of Harmonic Maps
Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
- 121
4
votes
0
answers
97
views
One-dimensional harmonic map flow with low regularity
My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in ...
- 9,853
1
vote
1
answer
420
views
Poincare Inequality on non compact manifold
Let $(X,g)$ be a non compact Riemannian manifold, such that its closure $\bar X=X\cup Y$ is a compact manifold with boundary $Y$.
Q: For the Poincare inequality
$$\|u\|_{L^2}\leq C \|\nabla u\|_{L^...
- 1,915
4
votes
0
answers
113
views
Are there any nontrivial examples of $C^1$ hypersurfaces with bounded (integrable) generalized mean curvature?
The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:
Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional ...
- 1,350
4
votes
0
answers
2k
views
What is the definition of "geometric analysis"? [closed]
Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone ...
- 49
3
votes
1
answer
454
views
How to evolve a star-shaped mean convex set to a strictly mean convex set?
I have trouble in going through a proof in a paper for quite a while. To simplify notation and make it readable to a larger audience, let me just present the simplest case:
Let $\Omega$ be a bounded ...
- 1,350
4
votes
0
answers
181
views
What is known for harmonic map flow in dimension > 2?
I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. ...
4
votes
1
answer
468
views
Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence
Thank you for spending time on the following question.
I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic ...
- 703
9
votes
1
answer
833
views
Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$
A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...
- 601
3
votes
2
answers
121
views
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
votes
1
answer
229
views
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
6
votes
0
answers
434
views
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
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
9
votes
3
answers
1k
views
Applications of geometric evolution equations.
Hi everybody,
I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological ...
10
votes
4
answers
1k
views
Finding constant curvature metrics on surfaces for the case of positive Euler characteristic
We approach the problem of finding a metric of constant curvature on a surface (i.e. a $C^\infty$ 2-manifold). Specifically, what we want to do is, given a surface $M$ and a metric $g_0$, show that ...
19
votes
3
answers
4k
views
Epsilon regularity: what does it say and where does it come from?
The $\varepsilon$-regularity phenomenon shows up in several different contexts. I try to describe it focussing on the harmonic map situation, but I really would like to understand the situation in ...
- 301