Questions tagged [geometric-analysis]

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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, $\...
zarathustra's user avatar
3 votes
1 answer
225 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 ...
gaoqiang's user avatar
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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 \...
Laithy's user avatar
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3 votes
1 answer
140 views

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 ...
William's user avatar
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3 votes
1 answer
127 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 ...
Leo Moos's user avatar
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3 votes
0 answers
91 views

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 ...
JMK's user avatar
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67 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. ...
Chris's user avatar
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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 ...
Leo Moos's user avatar
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2 votes
1 answer
129 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 ...
Leo Moos's user avatar
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5 votes
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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-...
Marc's user avatar
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Reference request: $ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $

I have a Riemannian manifold $M$ of dimension 2 on which I am considering the following equation: $$ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $$ on some patch $U$ of the manifold which is &...
Andreea M's user avatar
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2 votes
0 answers
82 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 ...
Cris.giansu's user avatar
2 votes
0 answers
65 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,...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
110 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-\...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
108 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 $\Sigma$ ...
Leo Moos's user avatar
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1 vote
0 answers
42 views

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 ...
Leo Moos's user avatar
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4 votes
0 answers
147 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 ...
Leo Moos's user avatar
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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 ...
Sven-Ole Behrend's user avatar
5 votes
0 answers
69 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, \...
Leo Moos's user avatar
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2 votes
0 answers
58 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 ...
Quanta's user avatar
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0 answers
113 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....
Leo Moos's user avatar
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2 votes
0 answers
123 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 ...
Leo Moos's user avatar
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7 votes
1 answer
282 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+...
Tree23's user avatar
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2 votes
0 answers
59 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 ...
Leo Moos's user avatar
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1 vote
0 answers
34 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 ...
Leo Moos's user avatar
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7 votes
0 answers
308 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 ...
Paul's user avatar
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0 answers
67 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 \...
Leo Moos's user avatar
  • 4,968
1 vote
0 answers
52 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{...
Leo Moos's user avatar
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4 votes
1 answer
158 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 ...
André Henriques's user avatar
2 votes
1 answer
95 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 ...
Leo Moos's user avatar
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3 votes
0 answers
88 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 ...
Leo Moos's user avatar
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2 votes
1 answer
210 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 $\...
gaoqiang's user avatar
  • 379
7 votes
2 answers
291 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 $u$. ...
gaoqiang's user avatar
  • 379
4 votes
0 answers
136 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 ...
Leo Moos's user avatar
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4 votes
1 answer
326 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{...
Leo Moos's user avatar
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2 votes
0 answers
144 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 ...
Leo Moos's user avatar
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2 votes
1 answer
174 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 ...
Shiyu's user avatar
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2 votes
0 answers
152 views

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,\...
Partha's user avatar
  • 759
2 votes
0 answers
514 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 ...
Nate River's user avatar
  • 4,802
3 votes
0 answers
156 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 ...
Partha's user avatar
  • 759
5 votes
0 answers
154 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 ...
maxematician's user avatar
2 votes
1 answer
62 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....
James Chiu's user avatar
4 votes
0 answers
107 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
6 votes
1 answer
278 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 $S^2$: $$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$ ...
Laithy's user avatar
  • 865
2 votes
0 answers
122 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
  • 865
3 votes
0 answers
157 views

How to show the upperbound of the Ricci tensor preserved on 3 manifold

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of ...
James Chiu's user avatar
5 votes
1 answer
180 views

3-manifolds with all minimal surfaces closed

Question. Let the manifold $(M^3,g)$ be compact without boundary. Suppose that every complete, embedded minimal surface $\Sigma \subset M^3$ is closed. Must $M$ be diffeomorphic to $\mathbf{S}^3$ or $\...
Leo Moos's user avatar
  • 4,968
4 votes
0 answers
143 views

What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?

As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
Leo Moos's user avatar
  • 4,968
3 votes
1 answer
388 views

Geometric flow by the level sets of a harmonic function

Let $u$ be an harmonic function in a cylindrical domain $B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma_t=\{u=t\}$ are graphs of functions on $B_2^{n-1}$. Consider a ...
Jingeon An-Lacroix's user avatar
1 vote
0 answers
186 views

What does Colding-Minicozzi theory say about convergence with multiplicity?

Let $(M_j \mid j \in \mathbf{N})$ be a sequence of compact minimal surfaces in the unit ball $B \subset \mathbf{R}^3$, with boundaries $\partial M_j \subset \partial B$. Assume that the sequence ...
Leo Moos's user avatar
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