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2
votes
3answers
294 views
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 ...
5
votes
1answer
140 views
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 ...
2
votes
2answers
86 views
Estimates on a heat process with fixed boundary data and zero initial conditions
Consider the following heat process:
For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that
$$
\partial_t ...
3
votes
0answers
54 views
Boundedness Spectral Triple Axioms for de Rham Complex
In Connes' axioms for a spectral triple $(A,H,D)$, they have a representation of an algebra $A$ in bounded operators on a Hilbert space $H$, and (unbounded) operator $D$, such that $[D,a]$ is bounded. ...
3
votes
1answer
99 views
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
votes
2answers
298 views
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 ...
6
votes
2answers
195 views
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. ...
2
votes
2answers
347 views
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
votes
1answer
151 views
Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R?
Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies ...
0
votes
0answers
34 views
Finding the lift of a curve under some assumptions
Let $f:\Omega\subseteq\mathbb{R}^n\to\mathbb{R}^n$ be a Lipschitz function and $h$ be a vector in $\Omega$. Assume that $0\in\Omega$ and $f(0) = 0$. Also, let $\sigma:[0,1]\to\mathbb{R}^n$ be the ...
5
votes
1answer
233 views
Green's function for *GJMS* operator
Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
0
votes
1answer
138 views
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. ...
2
votes
0answers
76 views
Deduce global estimate from scaling-invariant local estimate
Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
5
votes
0answers
279 views
Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
1
vote
1answer
119 views
Hausdorff measure and projections
Fix $ k \in \mathbb{N} $ and let $ H^k $ be the $k$-dimensional Hausdorff measure on $\ell^\infty $. Also, if $ V $ is a subspace of $ \ell^\infty $, we denote the projection onto $ V $ by $ \pi_V $. ...
3
votes
1answer
257 views
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$ ...
5
votes
1answer
325 views
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 ...
9
votes
3answers
286 views
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 ...
2
votes
0answers
31 views
Diameters of the images of two balls under a function
Let $ \Omega $ be an open and bounded subset of $ \mathbb{R^n} $, and let $ f:\Omega \to \mathbb{R} $ be a continuous function. I'm looking for some (preferably, minimal) conditions on $ f $ under ...
7
votes
1answer
229 views
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 ...
1
vote
3answers
315 views
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) ...
2
votes
2answers
215 views
Is there a combinatorial analogue of the Kazdan Warner theorem?
First let me state a result of Kazdan and Warner
Let $M$ be a compact orientable two dimensional manifold.
Let $f:M \rightarrow \mathbb{R}$ be a function that has the same
sign as the Euler ...
3
votes
0answers
176 views
What is known about analogous results of Kazdan and Warner in higher dimensions?
First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as ...
1
vote
1answer
517 views
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 ...
2
votes
1answer
153 views
Different forms of Bonnesen's strong isoperimetric inequality in the plane.
I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve:
$(R_{out} - R_{in})^2 \leq \pi^2 (L^2 - 4\pi A),$
where $R_{out}$ ...
10
votes
2answers
2k views
Surface equivalent of catenary curve
A catenary curve
is the shape taken by an idealized hanging chain or rope under the influence
of gravity. It has the equation $y= a \cosh (x/a)$.
My question is:
What is the shape taken by an ...
8
votes
3answers
717 views
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 ...
1
vote
1answer
386 views
What is the shape of a tight open trefoil?
Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically.
The problem is ...
3
votes
1answer
324 views
Geometric bound on the first eigenvalue of Laplace-Beltrami on forms
Let $M$ be a closed Riemannian manifold and $\Delta$ its Laplace-Beltrami operator. Then we have Yau's estimate on the first (non-zero) eigenvalue $\lambda_1>0$ of $\Delta$ acting on functions in ...
3
votes
2answers
655 views
Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature
A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...
9
votes
4answers
789 views
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 ...
13
votes
3answers
1k views
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 ...
5
votes
4answers
1k views
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 ...
