Questions tagged [geometric-analysis]

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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
1 answer
229 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-...
Marc's user avatar
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4 votes
0 answers
152 views

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
52 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
55 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
105 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
96 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
31 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
143 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
66 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
49 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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91 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
121 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
224 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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0 answers
54 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
  • 4,656
1 vote
0 answers
29 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
  • 4,656
7 votes
0 answers
248 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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66 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
  • 4,656
1 vote
0 answers
50 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
  • 4,656
4 votes
1 answer
151 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
93 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
82 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
183 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
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7 votes
2 answers
268 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
  • 129
4 votes
0 answers
104 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
316 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
139 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
0 answers
80 views

parabolic schwarz lemma

Trying to follow the computation in https://arxiv.org/pdf/math/0602150.pdf, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \text{tr}_{g}h = g^{i \bar l}...
Shiyu's user avatar
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2 votes
0 answers
147 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
  • 667
1 vote
0 answers
383 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
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3 votes
0 answers
155 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
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5 votes
0 answers
153 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
59 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
93 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
273 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
  • 721
2 votes
0 answers
118 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
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3 votes
0 answers
155 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
173 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
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4 votes
0 answers
140 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
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3 votes
1 answer
354 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
153 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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6 votes
1 answer
353 views

Which geometric variational problems admit an entropy identity?

Question. My question in broad terms: when and how can entropy be used in geometric variational contexts to study sequences of critical points, such as the two results described below? [See at the ...
Leo Moos's user avatar
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10 votes
1 answer
428 views

A counterexample to a conjecture of Lawson

Yau quotes Lawson as having formulated the following conjecture [1]: Let $M$ be an embedded minimal surface in $\mathbf{S}^3$. Prove that the two domains in $\mathbf{S}^3$ divided by $M$ have equal ...
Leo Moos's user avatar
  • 4,656
1 vote
1 answer
142 views

Minimal surfaces with increasing area but bounded Morse index

Question. What restrictions are known on closed manifolds $(M^3,g)$ that contain a sequence of embedded minimal surfaces $(\Sigma_j \mid j \in \mathbf{N})$ with \begin{equation} \mathrm{area} \, \...
Leo Moos's user avatar
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3 votes
0 answers
99 views

When is the least-area surface unique?

Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
Leo Moos's user avatar
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1 vote
0 answers
139 views

Can min-max be set up around a minimal cone?

Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom. Question. Given a regular minimal cone $\mathbf{C}$, can one set up a ...
Leo Moos's user avatar
  • 4,656
1 vote
0 answers
92 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 ...
Pete09's user avatar
  • 11
2 votes
0 answers
206 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 ...
Leo Moos's user avatar
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5 votes
0 answers
282 views

Are the two-valued homogeneous harmonic functions classified?

Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$? For reference, multi-valued functions are familiar objects in ...
Leo Moos's user avatar
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