All Questions
Tagged with geometric-measure-theory riemannian-geometry
49 questions
6
votes
2
answers
390
views
Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
3
votes
1
answer
189
views
Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
- 43
7
votes
2
answers
598
views
Invariance of the Lebesgue measure
It is well known that the Lebesgue measure is the unique (up to a multiplicative constant) sigma-finite Borel measure on $\mathbb{R}^d$ which is translation invariant.
I am wondering if a similar ...
- 401
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,155
1
vote
1
answer
181
views
For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point be of 0 volume measure
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}d\operatorname{vol}_g \forall E \in \mathcal{B}(M),$ the ...
- 1,467
2
votes
0
answers
187
views
Statistical invariants of Riemannian manifolds
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
- 159
2
votes
0
answers
91
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
- 125
1
vote
1
answer
160
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
3
votes
1
answer
349
views
In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
- 219
1
vote
1
answer
260
views
What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
- 305
2
votes
0
answers
207
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 ...
- 21
5
votes
0
answers
416
views
Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
- 16.6k
1
vote
1
answer
276
views
Defining area / n-volume of a finite metric space
Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested ...
- 820
3
votes
0
answers
100
views
Are there Lojasiewicz-Simon estimates with boundary?
Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.
Are there Lojasiewicz–Simon estimates ...
- 5,038
2
votes
0
answers
202
views
Geometric inequality related with convexity of the boundary
I'm new to Mathoverflow, so hopefully my question is well-posed.
My problem states as follows:
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...
- 21
9
votes
0
answers
202
views
approximation of currents
Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...
- 91
5
votes
0
answers
261
views
Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary
In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
- 409
1
vote
0
answers
134
views
Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
- 323
0
votes
0
answers
425
views
Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
2
votes
1
answer
741
views
Continuity of the perimeter of level sets w.r.t. level function
Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
- 1,759
2
votes
0
answers
354
views
Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
- 1,759
5
votes
2
answers
1k
views
Continuity of Hausdorff measure on level sets
Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that:
$\bullet$ $\phi^{-1}(0)\neq\emptyset$;
$\bullet$ $\nabla\phi(x)\...
- 1,759
1
vote
1
answer
258
views
Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
- 459
1
vote
1
answer
140
views
Isoperimetric inequality for exterior domains on $\mathbb{H}^{n}$
Fix $n\geq 2$ and let $\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$ be the hyperbolic space be defined as a Riemannian manifold equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}rd\...
- 459
2
votes
0
answers
144
views
Why are products of spheres integrable?
Let $n+1 \geq 3$ be an integer and $p + q = n$. Inside the unit sphere $\mathbf{S}^{n+1}$ the product
\begin{equation}
\mathbf{S}^p(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n}) = \{ (X,Y) \in \mathbf{R}...
- 5,038
1
vote
0
answers
157
views
Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
- 6,741
1
vote
1
answer
201
views
Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
0
votes
0
answers
131
views
Barycenters on Hadamard Manifolds
Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of ...
5
votes
0
answers
165
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 ...
- 5,038
6
votes
0
answers
279
views
Preiss' theorem on Riemannian manifolds
This may be a silly question as it seems one of the several Euclidean results in that carry over to Riemannian manifolds by passing in coordinates, but I suspect that the issue is subtler, and I am ...
- 3,223
4
votes
0
answers
125
views
An inequality about metric spaces
I started studying this article(《$L^2$ CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE》) about 3 months ago: arxiv.org/abs/1605.05583
In this article, there is a seemingly simple assertion ...
- 41
10
votes
0
answers
464
views
Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
- 7,817
6
votes
0
answers
249
views
Do asymptotically conformal maps converge to a weakly conformal map?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, ...
- 6,741
6
votes
0
answers
240
views
Minimizing area in relative homology class
A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...
- 2,095
2
votes
0
answers
92
views
lower volume bound of submanifolds with small mean curvature integral data
$(M^n,g)$ is a smooth submanifold in $\mathbb{R}^p$ ,and $B_1$ is the unit ball centered in the origin 0.
Is there a $\epsilon >0$，
when assuming $\int_{M\cap B_1} |H|^n \leq \epsilon$, and the ...
- 101
5
votes
0
answers
238
views
Is polar decomposition of a smooth map Sobolev?
Motivation:
Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
- 6,741
6
votes
0
answers
171
views
The distributional gradient of the closest isometry to the differential of a smooth map
The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
- 6,741
5
votes
1
answer
201
views
The Hausdorff dimension of the union of singular orbits and exceptional orbits
Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
user78128
6
votes
0
answers
156
views
Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?
It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even)
...
- 1,179
5
votes
1
answer
333
views
Second fundamental form blows up at minimal hypersurface singularity
I have seen the claim that "$A$ bounded on an area minimizing current implies no singular set" in a couple of papers by Lohkamp, but with no reference (see https://arxiv.org/abs/1805.02180 e.g.).
...
- 738
9
votes
0
answers
1k
views
Existence of barycenter
Let $(X,d)$ be a metric space. A barycenter of a Borel probability measure $\mu$ on $X$ is a minimizer of the function
\begin{equation}
\begin{split}
f \colon X & \to \mathbb{R}\\
x &\mapsto \...
- 1,347
2
votes
1
answer
316
views
Is the $L^p$ space of tensors complete?
On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm
$$||...
- 738
5
votes
1
answer
243
views
sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
- 49
14
votes
0
answers
632
views
Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
- 6,741
3
votes
0
answers
109
views
What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?
Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set $\...
- 1,771
1
vote
1
answer
463
views
Singularities in minimal surfaces [closed]
There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times ...
- 215
5
votes
1
answer
411
views
Averaging maps of Riemannian manifolds
Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon M\...
- 666
3
votes
0
answers
354
views
Brakke's theorem for gap in entropy between self-shrinkers
In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...
- 61
3
votes
1
answer
274
views
Compact Riemannian manifold with maximum average distance
Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...
- 3,017