All Questions
14 questions
3
votes
0
answers
157
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
$$h = O(1/r),\quad \...
- 969
5
votes
1
answer
310
views
Lee-Parker Yamabe problem proposition 4.6
I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
- 457
2
votes
0
answers
80
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
- 1,219
2
votes
0
answers
122
views
Local smoothness of harmonic heat flow
Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow
$$
\partial_tu-\...
- 1,219
2
votes
0
answers
134
views
Critical points of a strictly subharmonic function
Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant:
\begin{equation}
\Delta u = A > 0....
- 5,038
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
1
vote
0
answers
98
views
Gap phenomenon vs Rigidity results for surfaces
I am trying to understand the differences between the rigidity results and gap results for a given surface immersed into some manifold. For instance, a Gap theorem proved here (Theorem 2.7) says ...
- 11
2
votes
0
answers
208
views
Can you compute one eigenspace without computing them all?
Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as ...
- 5,038
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
5
votes
0
answers
165
views
Singularities of phase interfaces in closed surfaces
Let $(\Sigma,g)$ be a compact surface without boundary. Given $\epsilon > 0$, the $\epsilon$-Allen-Cahn equation is the semilinear elliptic PDE $\epsilon \Delta_g u - \epsilon^{-1} W'(u) = 0$, with ...
- 5,038
3
votes
1
answer
162
views
$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity
Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ ...
- 969
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
2
votes
0
answers
102
views
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 ...
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 $0$ ...
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