Questions tagged [geometric-analysis]
The geometric-analysis tag has no usage guidance.
82
questions
3
votes
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
$$
\lambda(\Sigma) := \sup_{x_0 \in \mathbb{R}^{n+1} \\ t_0 \...
2
votes
0answers
65 views
Reference question: $C^1$ estimate for a stable minimal surface
I am looking for an answer/reference to the following question: Let $(M,g)$ be a complete Riemannian manifold and $\Sigma\subset M$ a closed, stable minimal surface. Is it possible to prove $C^1$ ...
1
vote
0answers
119 views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
2answers
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
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 ...
4
votes
0answers
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
3answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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 ...
