# Questions tagged [geometric-analysis]

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82
questions

**3**

votes

**1**answer

131 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**

votes

**0**answers

70 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

**1**answer

141 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^...

**5**

votes

**1**answer

177 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**

votes

**0**answers

92 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 \...

**11**

votes

**2**answers

866 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

**1**answer

263 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

**0**answers

176 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 \...

**20**

votes

**2**answers

1k 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

**1**answer

182 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

**1**answer

493 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

**2**answers

163 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**

vote

**0**answers

71 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

**1**answer

102 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

**1**answer

288 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**

vote

**0**answers

41 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**

votes

**0**answers

69 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

**0**answers

65 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 $C^1$ ...

**1**

vote

**0**answers

119 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**

votes

**0**answers

146 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**

votes

**0**answers

66 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**

votes

**1**answer

501 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**

votes

**0**answers

96 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**

vote

**0**answers

154 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 ...

**3**

votes

**0**answers

83 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 ...

**2**

votes

**2**answers

245 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**

votes

**1**answer

366 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**

vote

**1**answer

290 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**

vote

**1**answer

82 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**

votes

**0**answers

339 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**

votes

**2**answers

593 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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**0**answers

96 views

### Are there any nontrivial examples of $C^1$ hypersurfaces with bounded (integrable) generalized mean curvature?

The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:
Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional ...

**46**

votes

**3**answers

2k views

### What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent ...

**4**

votes

**0**answers

1k views

### What is the definition of “geometric analysis”? [closed]

Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone ...

**2**

votes

**1**answer

81 views

### literature/reference request for estimates of first eigenvalue of certain Schrodinger operator on compact surfaces

On compact Riemannian surfaces (say without boundary), the Schrodinger operator I am interested in is of the form $-\Delta+2\kappa$, where $\kappa$ is the Gauss curvature. For minimal surfaces in $\...

**4**

votes

**0**answers

117 views

### Are heat kernels on metric measure spaces continuous?

Let $(M,d)$ be a separable, complete, compact metric space and $\mu$ a Radon measure with full support on it. Let $\mathcal{E}$ be a regular strongly local Dirichlet form on $L^2(M)$. There exists an ...

**2**

votes

**0**answers

158 views

### When is Cheng-Li-Yau heat kernel bound optimal?

In his Inventiones paper "The geometric complexity of special Lagrangian
$T^{2}$-cones" M. Haskins has used Cheng-Li-Yau theorem to give a lower bound of area of minimal Lagrangian tori in $\mathbb{CP}...

**3**

votes

**1**answer

246 views

### How to evolve a star-shaped mean convex set to a strictly mean convex set?

I have trouble in going through a proof in a paper for quite a while. To simplify notation and make it readable to a larger audience, let me just present the simplest case:
Let $\Omega$ be a bounded ...

**5**

votes

**1**answer

400 views

### Continuous deformation of soap films

Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?

**1**

vote

**1**answer

105 views

### Solvability of $u_t=\nabla_{u_\theta} u_\theta -\frac{1}{2}\nabla R +\frac{1}{2\tau}u_\theta + 2 Ric(u_\theta,\cdot)$

I ask a question in math.stackexchange, but nobody answer it. So, I ask here.
In Cao and Zhu's paper about Ricci flow, the $\mathcal{L}$-geodesic is defined as picture below.In fact, $\mathrm{Ric}(X,...

**3**

votes

**0**answers

100 views

### A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question
Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation:
$$
-\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0
$$
Here $a_i:...

**3**

votes

**0**answers

149 views

### What is known for harmonic map flow in dimension > 2?

I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. ...

**4**

votes

**1**answer

179 views

### Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$.
QUESTION I:
The above ...

**1**

vote

**1**answer

430 views

### Riemannian metric on a level set of a smooth function on a manifold

Also asked here: https://math.stackexchange.com/questions/1725491/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold
Let $(M,g)$ be a finite or infinite dimensional Riemannian ...

**3**

votes

**1**answer

276 views

### Riemannian Measures, Densities and Radon–Nikodym Theorem

If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int_M I_A \mu.
\end{equation}
My question ...

**2**

votes

**1**answer

511 views

### Sobolev Multiplication theorem for Fibre bundles

Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a smooth, Riemannian ...

**2**

votes

**1**answer

236 views

### How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$
with cone singularities along divisor
$D$
with cone angle $2\pi\beta$
is said to be
of
constant scalar curvature Kahler
or
cscK
if its scalar curvature $S(\omega)$, which is ...

**4**

votes

**2**answers

978 views

### Self-contained book on Ricci Flow/Geometric Analysis

Can someone please tell me whether there is any self-contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self-contained I mean it does not assume that ...

**3**

votes

**1**answer

343 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 ...