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### 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:...

**2**

votes

**0**answers

68 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

139 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

112 views

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

Also asked here: http://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

111 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

207 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

176 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

**1**answer

386 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

236 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 ...

**2**

votes

**3**answers

521 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

**1**answer

285 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 ...

**3**

votes

**2**answers

101 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 p(...

**3**

votes

**1**answer

196 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

**2**answers

377 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

**2**answers

243 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. ...

**3**

votes

**2**answers

550 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

**1**answer

167 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

**0**answers

39 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

**1**answer

274 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

**1**answer

185 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

**0**answers

79 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

**0**answers

301 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

**1**answer

149 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

**1**answer

320 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 $0$ ...

**6**

votes

**1**answer

364 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

**3**answers

326 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 ...

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votes

**0**answers

41 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

**1**answer

261 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

**3**answers

321 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) \...

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votes

**2**answers

245 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 ...

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votes

**0**answers

192 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 $\chi(M)$,...

**1**

vote

**1**answer

810 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

**1**answer

159 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

**2**answers

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 ...

**9**

votes

**3**answers

798 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

**1**answer

399 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

**1**answer

362 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

**2**answers

761 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

**4**answers

993 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

**3**answers

2k 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

**4**answers

2k 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 ...