# Tagged Questions

3answers
697 views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...
0answers
220 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
1answer
357 views

### Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function). I am looking for the solutions among of ...
1answer
253 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...
1answer
212 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...
1answer
141 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
1answer
288 views

### Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support. Is the reverse true? Namely that if some PDO ...
1answer
114 views

### Application of the inverse theorem function on singular points

This question is related to this one A question about the inverse theorem function in $\mathbb{R}^n$. In the response I received, I was told that the issue was related to the singularities, as I do ...
2answers
214 views

### A question about the inverse theorem function in $\mathbb{R}^n$

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ ...
0answers
88 views

### Problem reduced to analyzing solutions of a family of nonlinear systems of equations

I was able to reduce a research problem relating to normal numbers to analyzing the solutions of the following family of nonlinear systems of equations: \begin{align*} c_0+c_1+\cdots+c_{t-1}&=2t\\ ...
1answer
413 views

### Existence of Geodesics in continuous metrics

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...
0answers
208 views

### Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
1answer
450 views

### Counterexample to Sard Theorem for a not-C1 map

Is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is differentiable but not $C^{1}$, such that the image of the points where $f'(x) = 0$ has measure bigger than 0? If the answer is no, ...
1answer
756 views

### Are the only local minima of $\angle(v, Av)$ the eigenvectors?

Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle ...
1answer
341 views

### Nonnegative smooth function as sum of squares of smooth functions

There is a famous open problem, whose solution is attributed to Paul Cohen, but no published paper seems to be available: There exists $f\in C^\infty(\mathbb R,\mathbb R_+)$ such that $f$ is not a ...
1answer
462 views

1answer
336 views

### Regarding Discrete Eigenvalues

For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete. But, ...
1answer
470 views

### A differential inequality needed to prove a theorem about odd-dimensional souls

I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls: Suppose that ...
0answers
216 views

### glue together a sequence of holomorphic forms

hallo, my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...
1answer
285 views

### harmonic function on surface

Dear all, I am looking for reference books about real-valued harmonic functions on complete Riemannian surfaces, do you have any reference in your mind about this? I found some books about harmonic ...
6answers
6k views

### Why are there so many smooth functions?

I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it. Analycity imposes very strong conditions on a map, from ...
0answers
186 views

### Graph Laplacian to Continuous version

Has there been a study of vector laplacian that is a continuous version of a graph laplacian? Is there a good introduction to the topic?
1answer
628 views

### Is there a smooth map f:R^n --> R^{n-1} whose image separates the rationals and others?

Let me make the question more precise: Let A be the set of all points in R^n such that each of its coordinates is rational, and let B=R^n-A. My question is, is there a smooth map f:R^n --> R^{n-1} ...
0answers
267 views

### Differential forms on the simplex which are “constant towards the boundary”

Let $\Delta^k$ the standard $k$-dimensional simplex, $\Delta^k=\{(x^0,\dots,x^k)\in \mathbb{R}^{k+1}|\sum_{i=0}^kx^i=1\}$, and let $\Omega^\bullet(\Delta^k)$ be the de Rham complex of smooth ...
1answer
621 views

### diffeomorphic, holomorphic, biholomorphic

Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n-$dimensional manifold $X$. If we know that this mapping is diffeomorphic onto its ...
1answer
294 views

### convergence of metrics

Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$ ...
0answers
135 views

### Question about density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$ with $N$ not compact

Hi! Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. ...
3answers
1k views

### Analytical solutions of a differential equation (from Archimedes' Spiral)

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const. I've found that a) if $\phi \in (0,t)$, t is quite small, then $r(\phi) \approx k/2 *\phi^2$ b) if ...
3answers
632 views

### Hidden convexity

Suppose you are given a domain $\Omega \subset \mathbb{R}^n,$ and a (Morse) function $f: \Omega \rightarrow \mathbb{R},$ all of whose critical points are positive-definite. The question is: is there ...
0answers
418 views

### Analogue of Whitney's extension theorem

So let $n,m$ be two strictly positive numbers. Let $A\subseteq\mathbf{R}^n$ be a compact $C^{\infty}$-submanifold (possibly with boundary and of real dimension not necessarily equal to $n$). Let ...
1answer
325 views

### Smooth and analytic structures on low dimensional euclidian spaces

So it is relatively easy to show that there exists only one smooth structure on the real line $\mathbb{R}$. So here are 2 natural questions: Q1: Up to equivalence, is there only one real analytic ...
1answer
615 views

### Solution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk?

Hi, Perhaps it's a stupid question, in that case i'll delete it. Let M be a compact orientable smooth (Kahler if changes things) manifold of dimension $dim_{\mathbb{R}}(M)=2n$ with $n\geq1$, let ...
1answer
287 views

### question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$

Hi, i'm reading the proof of the fact that $C^{\infty}(M,N)$ is dense in the sobolev space $W^{1,m}(M,N)$, where $M,N$ are compact riemannian manifolds of dimension respectively $m,n$. I recall ...
1answer
203 views

### A question about to computing a integration

Let $M$ be a smooth manifold of 2n-dim, $v$ be a map from $M$ to the matrix of order $m\times m$. We call $p\in M$ is the singularity, if $v(p)$ is non-invertible. Suppose $v$ is smooth and the ...
1answer
566 views

### Reference for existence and uniqueness of differential equations for low differentiability?

My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and ...
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 ...
1answer
545 views

### Reference for a particular Radon transform on non-positively curved spaces

Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...
1answer
417 views