Questions tagged [geometric-analysis]
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166 questions
6
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2
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Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
0
votes
0
answers
65
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Regularity of Metric when defining C^k norms
Given $(M^n, g)$ closed riemannian manifold, I am wondering about the definition of the $C^k$ norms with respect to the metric, and how these norms depend on $g$. For example, I would assume that if $...
- 337
2
votes
0
answers
73
views
Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
- 337
14
votes
1
answer
573
views
Different proof techniques of the Atiyah-Singer index theorem
I am aware of the usual K-theoretical (cobordism, operator algebras) and heat kernel proofs of the index theorem, as answered in other questions in this site, e.g. here.
However, I recently read this ...
0
votes
0
answers
40
views
Is the principal eigenvalue of the clamped plate minimized by the equilateral triangle among all triangles of a given area?
Let $\Omega$ be a ``nice'' domain in $\mathbb{R}^2$
(bounded and with piecewise smooth boundary). Consider the clamped plate eigenvalue problem, given by
$$\Delta^2 u = \lambda u $$
$$ u|_{\partial \...
- 3,245
3
votes
0
answers
68
views
How powerful are sequences of Steiner symmetrizations?
I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
- 171
1
vote
0
answers
56
views
Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
- 383
5
votes
0
answers
338
views
Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions
Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
- 319
3
votes
1
answer
286
views
A compact Riemann surface with a finite set of points removed is parabolic
A Riemann surface $\mathcal{R}$ is called parabolic if it is not compact and doesn't carry a negative non-constant subharmonic function, and is called hyperbolic if it carries a negative non-constant ...
- 438
3
votes
0
answers
159
views
Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
- 969
3
votes
1
answer
152
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Derive distributional inequalities from pointwise estimates
My question is how to prove the following claim:
Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous
on $\mathbb{R}^n$. If ...
- 33
4
votes
1
answer
181
views
Is every area-minimizing cone a level set of a least-gradient function?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
- 5,048
3
votes
0
answers
100
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Asymmetric minimal surfaces in $H^3$
Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 ...
- 337
5
votes
0
answers
104
views
Is it always possible to find a conjugate optical function?
Optical functions (functions with null gradients) and double null foliations (foliations of a spacetime by two related optical functions) play a large roll in modern mathematical relativity research. ...
- 419
1
vote
0
answers
67
views
Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
- 5,048
2
votes
1
answer
148
views
Minimal graph with confusing (?) property
Let $n \geq 2$ and $C = \{ (x,y) \in \mathbf{R}^{2n} \mid \lvert x \rvert = \lvert y \rvert \} \subset \mathbf{R}^{2n}$ be the Simons cone. (Whether this is area-minimizing or not does not seem to ...
- 5,048
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
207
views
Tangent cones at infinity and the regularity of minimal submanifolds
In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
- 21
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
1
vote
1
answer
123
views
Singularities of mean-convex MCF in the sphere?
Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...
- 5,048
2
votes
1
answer
151
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A question related to Ros's two-piece property
In 1995 Ros proved that minimal surfaces in the round three-sphere $S^3$ enjoy a two-piece property: any hyperplane $\Pi \subset \mathbf{R}^4$ divides every minimal surface $\Sigma \subset S^3$ into ...
- 5,048
4
votes
0
answers
153
views
Is there a good description of harmonic maps from $\mathbf{C}$ to $\mathbf{H}$?
Given a non-constant holomorphic quadratic differential $\phi \, \mathrm{d} z^2$ on the complex plane $\mathbf{C}$, there is a harmonic diffeomorphism $u: \mathbf{C} \to \mathbf{H}$ into the ...
- 5,048
1
vote
0
answers
61
views
Existence of solution to prescribed curvature problem with given asymptotic on the punctured unit disc
I have trouble understanding a conclusion in the following paper: Prescribed Curvature and Singularities of Conformal Metrics on Riemann Surfaces by Robert C. McOwen
In the appendix, part B, we are ...
5
votes
1
answer
147
views
What is the Morse index of the Scherk surfaces?
The Scherk surfaces are properly embedded, complete minimal surfaces
\begin{equation}
S_\alpha \subset \mathbf{R}^3
\end{equation}
that are asymptotic at infinity to the union of two planes $\Pi_1, \...
- 5,048
2
votes
0
answers
71
views
Examples of elementary group of isometries of the ideal boundary of hyperbolic plane
A Riemannian manifold $(X,g)$ is called Hadamard manifold if it is complete and simply connected and has everywhere non-positive sectional curvature. An example of such a manifold would be the ...
- 41
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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....
- 5,048
2
votes
0
answers
134
views
What prevents spontaneous oscillations in minimal surfaces?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be an unstable minimal cone with an isolated singularity at the origin. Let $\Sigma \subset \partial B$ be its link, and $(\varphi_i)$ be the eigenfunctions ...
- 5,048
7
votes
1
answer
511
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
2
votes
0
answers
65
views
Defining minimality 'through deformations'
Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
- 5,048
1
vote
0
answers
40
views
Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 5,048
8
votes
0
answers
370
views
Existence of Yang-Mills connection
My question is about what we know, in dimension $4$, about the loss of compactness of Yang–Mills connections with $L^2$-bounded curvature. My background is more analytical than geometrical and it is ...
- 914
2
votes
0
answers
79
views
Curvature estimate in terms of the boundary
The curvature of a minimal disc $S^2 \subset \mathbf{R}^3$ can be bounded in terms of the curvature of its boundary via the Gauss–Bonnet formula:
\begin{equation}
\frac{1}{2}\int_S \lvert A \rvert^2 \...
- 5,048
1
vote
0
answers
59
views
Dirichlet-to-Neumann estimate for minimal graphs
Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{...
- 5,048
4
votes
1
answer
181
views
One-sided version of the curve-shortening flow
The curve-shortening flow is
$$
\frac{\partial C}{\partial t} = \kappa n
$$
where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed ...
- 43.2k
2
votes
1
answer
98
views
The attractive 'force' between phase interfaces in the Allen-Cahn model
The heuristic explanation of the behavior of phase transition in the Allen–Cahn model describes two 'forces' at play:
the curvature of the phase interfaces—they each 'want to' minimize length;
and an ...
- 5,048
3
votes
0
answers
101
views
Are there Lojasiewicz-Simon estimates with boundary?
Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.
Are there Lojasiewicz–Simon estimates ...
- 5,048
2
votes
1
answer
276
views
The space of Sobolev maps between Riemannian manifolds
Let $\mathcal{M}, \mathcal{N}$ be two Riemannian manifods. Suppose that $\mathcal{N}$ is properly and isometrically embedded in $\mathbb{R}^n$. The space of Sobolev maps between $\mathcal{M}$ and $\...
- 438
7
votes
2
answers
368
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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
4
votes
0
answers
193
views
What are the next-simplest area-minimizing cones?
The simplest area-minimizing, codimension one cones $\mathbf{C} \subset \mathbf{R}^{n+1}$ are the Simons cones. I am trying to understand the behavior of area-minimizing cones a bit better, but these ...
- 5,048
4
votes
1
answer
346
views
Is there a harmonic function with just one singular point?
Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy
\begin{equation}
\begin{cases}
\Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\
\frac{...
- 5,048
2
votes
0
answers
175
views
When are the Schoen-Yau minimal surfaces embedded?
In 1979, Rick Schoen and Shing-Tung Yau published an Annals paper in which they proved the existence of so-called 'incompressible' minimal surfaces in compact Riemannian manifolds.
Question. Under ...
- 5,048
2
votes
1
answer
181
views
Parabolic Schwarz lemma
Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
- 59
2
votes
0
answers
153
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An equality regarding Dirac operator
Let S be a spinor bundle on a closed Riemannian manifold M, with a spin connection A. Then for a spinor field $\phi$, we know
\begin{align*}
\frac{1}{2}\Delta|\phi|^2=\langle \nabla_A^*\nabla_A \phi,\...
- 954
2
votes
0
answers
665
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 ...
3
votes
0
answers
172
views
A pde inequality
Say $M$ be a closed manifold of dimension $6$, we have
\begin{align*}
\Delta f\leq g f-f^2
\end{align*}
where $g$ is a smooth function on $M$ and $f\geq 0$ (in my case $f=|\phi|^2$ for $\phi$ a ...
- 954
5
votes
0
answers
160
views
Potential theory as a tool in extrinsic flows
Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
- 217
2
votes
1
answer
68
views
Question of an inequality from curve-shortening flow
I am reading a paper about the curve shortening flow which make use of one inequality but I don't know where does it come from where f(x,t) is a smooth function and C is a constant depending on time t....
- 273
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
6
votes
1
answer
296
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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