Questions tagged [geometric-analysis]
The geometric-analysis tag has no usage guidance.
166 questions
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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 \...
- 823
4
votes
0
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110
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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-...
- 419
1
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0
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106
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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 ...
- 5,146
5
votes
1
answer
412
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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?
- 51
3
votes
1
answer
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$\Delta_g f = 0$ 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 $\Delta_g f = 0$ ...
- 969
3
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2
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452
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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 ...
- 381
5
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2
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1k
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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 ...
- 789
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3
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Epsilon regularity: what does it say and where does it come from?
The $\varepsilon$-regularity phenomenon shows up in several different contexts. I try to describe it focussing on the harmonic map situation, but I really would like to understand the situation in ...
- 301
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4
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Finding constant curvature metrics on surfaces for the case of positive Euler characteristic
We approach the problem of finding a metric of constant curvature on a surface (i.e. a $C^\infty$ 2-manifold). Specifically, what we want to do is, given a surface $M$ and a metric $g_0$, show that ...
4
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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 ...
- 49
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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 ...
- 121
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1
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420
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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
$$\|u\|_{L^2}\leq C \|\nabla u\|_{L^...
- 1,915
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0
answers
126
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On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\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\...
- 169
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1
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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 ...
- 1,467
1
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0
answers
84
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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 ...
- 381
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4
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How does curvature change under perturbations of a Riemannian metric?
Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
- 8,532
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0
answers
84
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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 \...
- 823
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3
answers
2k
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What are Carnot groups?
I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
- 520
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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$ ...
- 308
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785
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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 ...
- 799
3
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1
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455
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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 ...
- 1,350
2
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2
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993
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Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps?
Let $(N,J_N)$ and $(M, J_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J_N$ and $J_M$).
Does it imply ...
- 3,245
2
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0
answers
185
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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 ...
4
votes
0
answers
89
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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 ...
- 1,630
1
vote
0
answers
268
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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 ...
- 623
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2
answers
656
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Behavior of the spectrum of the Laplacian under pointed smooth convergence
The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$.
On the other ...
- 4,304
2
votes
0
answers
187
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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}...
user74900
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3
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Applications of geometric evolution equations.
Hi everybody,
I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological ...
9
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1
answer
833
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Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$
A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...
- 601
1
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3
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Estimating L1 functions over the ball with radius 2r
Let $ f $ be in $ L^1(\Omega) $ where $
\Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such that $ B(x_i,2r_i) \...
- 520
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3
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Duality relations for Lebesgue spaces of sections of vector bundles
Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle_x$ on the ...
user14166
4
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1
answer
620
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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 ...
- 5,405
2
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1
answer
667
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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 ...
- 303
1
vote
1
answer
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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$ ...
user101142
6
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2
answers
554
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Hamilton-Ivey pinching in dimension 4
I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. ...
- 787
4
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0
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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 ...
- 9,863
4
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0
answers
192
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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 ...
- 687
1
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1
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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,...
- 165
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1
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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 ...
- 703
4
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0
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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 ...
- 1,350
4
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1
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197
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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 ...
- 1,467
4
votes
1
answer
686
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Horizontal lift of differential operator
On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a ...
- 9,863
2
votes
1
answer
95
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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 $\...
- 783
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1
answer
699
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Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?
On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
user14166
7
votes
1
answer
785
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Influence of Yau's solution to the Calabi Conjecture on the field of PDEs
I remember reading a long time ago(I can't recall where, unfortunately) that Yau's solution of the Calabi-Yau conjecture introduced new techniques that were very important for the field of partial ...
- 1,665
2
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1
answer
266
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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 ...
user21574
4
votes
0
answers
182
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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. ...
2
votes
1
answer
2k
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Vitali Covering Theorem for Arbitrary Subsets of Doubling Metric Spaces
Hello,
I've been wondering today around the following exercise in J.Heinonen's "Lectures on Analysis on Metric Spaces" (see below for terminology): prove that the statement of Vitali Covering Theorem ...
- 486
1
vote
1
answer
525
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Hamilton's Proof of the Tensor Maximum Principle
My questions come from Richard Hamilton's Three-Manifolds with Positive Ricci Curvature paper. I'm trying to work through parts of the paper so I can better understand the Ricci Flow for my research. ...
- 787
7
votes
1
answer
555
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Lower bound on $L^2$ norm of mean curvature in general dimensions
Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$
$$
\int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|}
$$
by Gauss-Bonnet and that this is saturated ...
- 2,299