1
vote
1answer
354 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 ...
1
vote
1answer
244 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 ...
3
votes
1answer
209 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 ...
2
votes
1answer
140 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 ...
4
votes
1answer
282 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 ...
0
votes
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 ...
0
votes
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 $j(f)(x)$ ...
1
vote
0answers
82 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\\ ...
2
votes
1answer
409 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 ...
3
votes
0answers
203 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 ...
14
votes
1answer
448 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, ...
19
votes
1answer
754 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 ...
13
votes
1answer
340 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 ...
1
vote
1answer
461 views

Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...
2
votes
3answers
438 views

How do we use an Ehresmann connection to define a semispray?

Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle. Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...
7
votes
1answer
203 views

Extremals versus minima for variational problems

A geodesic on a Riemannian manifold is generally not the shortest among nearby curves with the same endoints. But it can always be divided into parts each the shortest among nearby curves between its ...
2
votes
0answers
130 views

Extrapolation on the p-norm sphere using the exponential map

Hi, In order to follow a branch of solutions to an implicit equation on the manifold $M = \lbrace x \in \mathbb R^n, \|x\|_p = 1 \rbrace $, I'm interested in the following problem. Given two points ...
6
votes
1answer
357 views

How “should” I define “absolutely continuous” functions on e.g. n-spheres?

(Am writing this post in a rush, out of office, so cannot give adequate links etc right now.) There is a classical and well-understood definition of what it means for a continuous function ...
6
votes
2answers
598 views

Linearly independent vector fields

Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$. Question: Is it possible to choose a cube $C$ around ...
4
votes
1answer
629 views

Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading: What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
-1
votes
1answer
280 views

Global invertibility

A differentiable transformation of R^n at each point has an invertible derivative. Does it imply that the transformation is a global diffeomorphism?
6
votes
1answer
542 views

On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?

I could not answer or find references of this question, even for the following special case: On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function ...
4
votes
2answers
623 views

gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail? Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ...
3
votes
1answer
334 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, ...
8
votes
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 ...
1
vote
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, ...
2
votes
1answer
283 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 ...
25
votes
6answers
5k 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 ...
1
vote
0answers
185 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?
4
votes
1answer
627 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} ...
5
votes
0answers
265 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 ...
4
votes
1answer
616 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 ...
0
votes
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 $m\geq 0$ ...
4
votes
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 $W^{1,p}(M,N)$. ...
4
votes
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 ...
6
votes
3answers
631 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 ...
0
votes
0answers
416 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 ...
6
votes
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 ...
4
votes
1answer
613 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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
1answer
564 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 ...
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 ...
11
votes
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 ...
6
votes
1answer
417 views

Estimating the flow when we know the vector field

Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...
22
votes
2answers
2k views

Examples of loss of regularity by “creation of topology”

I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem ...
6
votes
2answers
424 views

Can I detect the point of impact without looking at it?

I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...
92
votes
16answers
12k views

How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
21
votes
7answers
2k views

Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be. "Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that F(a)=F(b) and F'(t) exists ...
17
votes
5answers
2k views

Sheaves and Differential Equations

How do sheaves arise in studying solutions to ordinary differential equations? EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to ...