Questions tagged [geometric-analysis]

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5
votes
1answer
267 views

Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall ...
1
vote
1answer
91 views

Asymptotics of constant mean curvature surfaces

Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$ In the case where the dimension is $n = 2$, $\Sigma$ is non-...
5
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0answers
79 views

Minimal cones and homology spheres

Let $\Sigma \subset \mathbf{S}^{n}$ be a codimension one, embedded minimal surface in the round $n$-dimensional sphere. Let moreover $\mathbf{C} = \mathbf{C}(\Sigma)$ be the minimal cone in $\mathbf{R}...
1
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1answer
97 views

How can I check this volume comparison?

I am reading the paper Ricci Curvature and Volume Convergence written by Professor Colding. In section 2, they define Lipschitz functions $b_j^+:M\to\mathbb R$ with $|\nabla b_j^+|=1$ and set $$\Phi=(...
5
votes
1answer
181 views

Gradient of solution to heat equation under evolving metric

The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
3
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0answers
52 views

Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$

Let $\gamma$ be a metric on $S^2$. I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$: $$div_{\gamma} A = \omega$$ where $\omega$ is a 1-form. It is known that there ...
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0answers
88 views

Mean Value Inequality with Linear Term

I am having trouble proving this modified mean-value inequality. Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$ Prove that there exists constants $r_0,C>0$ depending only on $c$...
3
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0answers
81 views

Calculation of the mean curvature under a normal perturbation

Let $X: M^n \to N^{n+1}$ be a Riemannian immersion. Write $g, A, \nu, H$ for the first fundamental form, second fundamental form, Gauss map and mean curvature of $X$ respectively. Consider the normal ...
2
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0answers
81 views

For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $...
5
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0answers
113 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 ...
3
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0answers
141 views

Dirichlet to Neumann operator and the Riesz transform

Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$....
5
votes
1answer
214 views

A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues

Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda_i$ of $-\...
2
votes
0answers
233 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(\...
4
votes
1answer
168 views

Neckpinch singularity of Ricci flow

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the ...
0
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0answers
72 views

A calculation involving Cotton tensor

I have a confusion regarding a calculation given below : $$ \begin{split} \int_M C^{ij}\nabla_i f \nabla_j f d\mu & = \frac{1}{3}\int_M g^{ij}g_{ij} C^{ij}\nabla_i f \nabla_j f d\mu \\ &= \...
16
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0answers
596 views

Schoen and Yau's proof of the higher dimensional positive mass theorem

In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July ...
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0answers
51 views

Modelling fluid flows with mean curvature flow

A while ago I was wondering if the displacement of fluid described in this blog post could be modelled with mean curvature flow or some other flow, but when I asked someone in Engineering they replied ...
3
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0answers
141 views

Reference on noncommutative PDE

I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
4
votes
1answer
193 views

Ricci flow proof of isoperimetric inequality

It is well-known in geometric analysis that one can use curve-shortening flow to prove the isoperimetric inequality (where the general result requires curve-shortening flow for non-convex curves). I ...
3
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0answers
81 views

Stability of bubbles under the heat flow

Let $\Phi : S \times [0,\infty) \to M$ be Struwe's weak global solution to the heat equation with smooth initial data $\phi : S \to M$, where $S$ is a compact surface and $M$ is a compact three-...
1
vote
1answer
157 views

Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$

Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with \begin{align} 0<\int_{\Sigma}f^...
6
votes
1answer
221 views

Completeness hypothesis in the positive mass theorem

I am trying to understand and further formalize Witten's proof of the positive mass theorem. Dan Lee, in his book "Geometric relativity" did a wonderful job with formalizing and carrying out the ...
4
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0answers
95 views

Characterization of geodesic balls

In $\mathbb R^{n\geq3}$, spheres can be characterized by single layer potentials having constant eigenfunction. More precisely we have the following: Theorem (H. Shahgholian) Let $\Omega\subset \...
12
votes
2answers
1k views

Is there a solution of the Yamabe problem using Ricci flow?

Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in ...
7
votes
1answer
303 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-...
5
votes
0answers
210 views

On Colding-Minicozzi limit lamination theorem

Colding and Minicozzi proved the following limit lamination theorem (see Theorem 8.26 in "A course in Minimal Surfaces" by Colding and Minicozzi for the complete statement) THEOREM: Let $\Sigma_i \...
21
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2answers
2k views

Formal mathematical definition of renormalization group flow

I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what ...
7
votes
1answer
207 views

Path of Diffeomorphisms Fixing the Boundary

Could you please let me know if the following is true. The problem came up while constructing a solution of a PDE. I have browsed through the net for an answer. While I came across some articles ...
11
votes
1answer
525 views

Are there currently any plausible approaches to proving the Penrose сonjecture?

I have recently been reading some of the literature on the Penrose inequality, especially the papers by Bray and by Huisken and Ilmanen. One notices immediately that the existing proofs for the ...
3
votes
2answers
226 views

Proof of Isoperimetric Inequality using Curve Shortening Flow

I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...
1
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0answers
72 views

Existence of nonparabolic ends

Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...
3
votes
1answer
126 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 $\Delta_g f = 0$ ...
5
votes
1answer
317 views

Volume comparison on Grassmannian

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
1
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0answers
59 views

Stable region of minimal hypersurfaces with finite Morse index

In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1): Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
2
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0answers
72 views

Is Colding-Minicozzi entropy continuous w.r.t. $C^\infty$ convergenge?

For an hypersurface $\Sigma^n \subseteq \mathbb{R}^{n+1}$ the entropy introduced by Colding and Minicozzi (see their paper) is defined as $$ \lambda(\Sigma) := \sup_{x_0 \in \mathbb{R}^{n+1} \\ t_0 \...
2
votes
0answers
69 views

Reference question: $C^1$ estimate for a stable minimal surface

I am looking for an answer/reference to the following question: Let $(M,g)$ be a complete Riemannian manifold and $\Sigma\subset M$ a closed, stable minimal surface. Is it possible to prove $C^1$ ...
2
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0answers
122 views

On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define $$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$ where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
6
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0answers
152 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 ...
4
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0answers
74 views

Morse index of a closed minimal surface with a small disc removed

Consider the following observation: Let $c_1$ be a geodesic on the unit round sphere $S^2$ with length $2\pi-\epsilon$, where $\epsilon$ is sufficiently small. Then $c_1$ has Morse index one ...
12
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1answer
552 views

Sheaf-theoretically characterize a Riemannian structure?

A smooth structure on a topological manifold can be characterized as a sheaf of local rings, see for example the discussion here. Q: Is there a way to characterize a Riemannian structure on a smooth ...
2
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0answers
108 views

Product of cotton-york tensor with ricci tensor

In the process of calculation of a problem in tensor analysis, I have encountered with an expression given by $$C^{ij}R_{ij}=0.$$ This is for a Riemannian manifold of dimension 3. Assuming the ...
1
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0answers
170 views

Maximum principle on noncompact manifold with boundary

On a complete noncompact manifold with ricci curvature bounded below, we have Yau's generalized maximum principle. What if we have we have noncompact manifold with compact boundary and one complete ...
4
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0answers
86 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 ...
3
votes
2answers
272 views

Resources on Elliptic Boundary Value Problems on manifolds

My situation: I am currently trying to understand Uhlenbecks results on the Yang Mills equation. One of the most common notions in this paper is that of an elliptic system or an elliptic boundary ...
11
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1answer
405 views

Smallest volume representatives of homology

Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when ...
1
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1answer
311 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
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1answer
92 views

If the first Dirichlet eigenfunction on a set $D$ is regular up to the boundary, is $D$ regular?

Given any open set $D$ in $\mathbb R^n$, we can define the first Dirichlet eigenfunction $u$ of $-\Delta$ on $D$ as the minimizer of the Rayleigh quotient over $H_0^1(D)$. Interior regularity of $u$ ...
4
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0answers
398 views

the regularity of the level set of distance function on Riemannian manifolds

On a complete smooth Riemannian manifold $(M^n, g)$, define $f(x)= d(p, x)$ where $p, x\in M^n$, and $d$ is the distance determined by the Riemannian metric $g$. Do we have: Except a set with zero ...
10
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2answers
647 views

Non-compact manifolds of positive/non-negative Ricci curvature

Consider a non-compact complete Riemannian manifold $(M, g)$ with smooth compact boundary $\partial M$. Suppose also that $M \setminus \partial M$ has positive/non-negative Ricci curvature. My ...
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0answers
48 views

In a short time $0<t<T$, does the $C^{1,\alpha}$ norm of mean curvature flow depend only on $T$ and the $C^{1,\alpha}$ norm of initial surface?

Given a smooth compactly embedded initial hypersurface $M_0$ and let $M_t$ ($0<t<T$) be a smooth solution of mean curvature flow starting from $M_0$. Then how does the regularity of $M_t$ depend ...