All Questions
Tagged with dg.differential-geometry real-analysis
158 questions
8
votes
0
answers
103
views
Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
- 1,429
6
votes
1
answer
197
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 1,429
2
votes
2
answers
158
views
Does there exist a continuous field of directions in $\mathbb R^3$ tangent to every sphere?
Does there exist a nonconstant continuous map $v: \mathbb R^3 \to \mathbb S^2$ such that every sphere $S \subset \mathbb R^3$ is tangent to $v(x)$ at some $x \in S$?
Bonus: I also suspect that for ...
- 6,215
4
votes
1
answer
128
views
Lower bound of mean curvature implies that the set is subset of a given ball
If a simply connected set $\Omega\subset\mathbb{R}^n$ has $C^2$ boundary such that the mean curvature $H$ of $\partial \Omega$ satisfies:
$$H\geq 1$$
Does this imply that $\Omega\subset B_1$ after ...
- 421
5
votes
1
answer
374
views
What is the length of an algebraic curve?
The following question seems to be somewhat standard, but I was unable to find any reference. I would be grateful for any pointers to relevant literature.
We consider a real polynomial $p(x,y)$ of ...
- 53
5
votes
1
answer
196
views
What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?
Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
- 641
1
vote
0
answers
54
views
Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
2
votes
1
answer
146
views
Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane
Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...
- 7,527
7
votes
2
answers
627
views
Elliptic regularity on manifolds: Is this true?
Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
- 1,171
0
votes
0
answers
126
views
A question about associated operator on continuous functions space equiped with L2 norm
For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
- 39
1
vote
1
answer
160
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
2
votes
1
answer
138
views
Boundedness of an exit time from a campact set
Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\begin{align*}
& x(0)=x_0 \\
& \dot{x}=v(x).
\end{...
- 449
0
votes
0
answers
97
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
1
vote
1
answer
345
views
Topological degree of differentiable map using line integrals?
Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
12
votes
4
answers
1k
views
Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$
I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality:
$$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$
where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
- 7,527
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
1
vote
0
answers
155
views
Does there always exist a regular curve connecting two points in an open connected subset of $\mathbb{R}^n$? [closed]
As the title says, given $A\subseteq \mathbb{R}^n$ open and connected and $x, y\in A$, I am looking for a continuous curve $\gamma:[0, 1]\rightarrow A$ which is differentiable in $(0,1)$ with $\gamma'(...
- 21
2
votes
0
answers
128
views
Making a continuous function into embedding by adding additional dimension
While doing my researches, I encountered the following problem.
Let $f:[0,1]^n\rightarrow \mathbb{R}^{n+k}$ be an arbitrary continuous function.
I want to make this function an embedding by perturbing ...
- 173
2
votes
1
answer
181
views
Matrix function as gradient
Let $S_n^{++}(\mathbb{R})$ be the space of $n \times n$ symmetric positive definite matrices. For $M \in S_n^{++}(\mathbb{R})$ consider the function $f: X \in S_n^{++}(\mathbb{R}) \mapsto M X^{-1} M$.
...
- 407
0
votes
0
answers
75
views
Morse functions on subset $\bar \Omega$ of $\mathbb {R}^d$ and its level sets
Let $f$ be a $C^{2}(\bar{\Omega})$ Morse function, where $\Omega$ is a bounded open set of $\mathbb{R}^d$: this means that
$$
\begin{cases}
f(x) = 0 \\
\nabla f (x) \neq 0
\end{cases}\text{ on }\...
- 61
4
votes
1
answer
202
views
Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
- 4,145
2
votes
1
answer
232
views
Existence of diffeomorphism interpolating affine map and identity
$\newcommand{\R}{\mathbb{R}}$Suppose $\Omega$ is a bounded, convex domain in $\R^{m}$. Fix $x_1, x_2\in\Omega$ and an invertible matrix $A\in\mathrm{GL}^{+}(m)$ with positive determinant.
Let $U\...
- 193
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
1
vote
0
answers
131
views
Integral flow that can commute to Laplacian operator
Firstly, considering the vector field in $ \mathbb{R}^3 $, $ X=x_2e_1-x_1e_2 $, we can see that
$$
\phi(t,x)=\phi(t,x_1,x_2,x_3)=(t,x_1\cos t+x_2\sin t,-x_1\sin t+x_2\cos t,x_3)
$$
is the ...
- 1,219
2
votes
0
answers
106
views
Approximating PL homeomorphism by diffeomorphisms in Euclidean space
The question is whether a piecewise function can be approximated by diffeomorphic functions in the following two situations.
I'm not really familiar with these piecewise stuffs. So it may be stupid ...
- 173
0
votes
0
answers
155
views
Implicit function theorem on curves
I am trying to figure out, whether the IFT can be generalized to curves. Let's say I have a function $G(x,u)$ mapping $\mathbb{R}^{n+m}\rightarrow \mathbb{R}^n$ with invertible jacobian $\frac{\...
0
votes
0
answers
168
views
How does one make sense of singular solutions to constant mean curvature equation?
Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
- 537
2
votes
1
answer
150
views
Continuity of a reaching time of a submanifold
Let $\mathcal{O}$ be a bounded open subset of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(\mathcal{O},\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\...
- 449
3
votes
1
answer
340
views
Shrinking a disk with fixed differential
Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{...
- 134
3
votes
1
answer
198
views
Volume of 3-dimensional region
Let $G$ be bounded finitely connected domain in $\mathbb{R}^3$ with 2-smooth boundary $\partial G$ each connected component of which has positive Gaussian curvature.
Each sufficiently small open ...
- 197
0
votes
2
answers
630
views
Smooth approximation for non differentiable function
Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any ...
- 3,625
2
votes
0
answers
95
views
An inequality in Huisken paper
I am reading a paper is written by Gerhard Huisken 'Flow by mean curvature of convex surfaces into sphere' and I want to show the following statement
Let $p\ge 2$ then for $\eta>0$ and any $0\le \...
- 273
4
votes
1
answer
407
views
Lipschitz-regularity of partition of unity
Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
- 195
0
votes
0
answers
76
views
Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998
1
vote
0
answers
46
views
Help with a surface of delay differential equations
This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
- 388
2
votes
1
answer
136
views
Expressing a vector valued function in terms of its derivatives
Consider a function
$$
f:\mathbb{R}^n\rightarrow\mathbb{R}^m
$$
given by $m$ functions $f_i:\mathbb{R}^n\rightarrow \mathbb{R}$ that we can assume to be polynomials in $x_1,\dots,x_n$.
Does there ...
- 23
5
votes
1
answer
207
views
The Lipschitz constant of convex sphere in $\mathbb{R}^3$
Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, admitting a bijective map to the unit round ...
- 799
3
votes
1
answer
198
views
Smoothness of ruled surface (asymptotic) parameterisations
A ruled surface $S$ shall be defined as surface consisting of straight line segments. It is commonly known (cf. [BER, p.362] or [STR, p.93] - bibliography at the end) that a ruled surface allows for a ...
11
votes
1
answer
452
views
Does every smooth map of rank at most d factor through a d-manifold?
Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...
- 37.8k
2
votes
0
answers
148
views
Finding an asymptotically flat manifold with ${\rm Ric}_{r\phi} = \frac{\sin\theta}{r^2}$
Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$.
Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1$...
- 969
1
vote
0
answers
159
views
Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
1
vote
1
answer
137
views
Smoothness of the asymptotic parametrization of a ruled surface
Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form
$$\begin{align}
\sigma \colon I \...
- 985
3
votes
0
answers
200
views
The best applications of the Poincaré-Bendixson theorem [closed]
I'm reading about the Poincaré-Bendixson theorem in the plane, I really liked the theorem. I have seen common applications in Sotomayor and Perko's book. But I would like to know what other ...
- 143
3
votes
0
answers
185
views
Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
- 11.2k
0
votes
1
answer
711
views
Lipschitz domains ambiguous definitions
I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...
- 1,759
1
vote
0
answers
259
views
Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
5
votes
1
answer
339
views
Finding vector fields on $S^2$ with equal divergence
Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal ...
- 969
7
votes
1
answer
246
views
Currents in sub-Riemannian geometry
Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The ...
- 215
9
votes
2
answers
777
views
Can the thief escape (from a smooth, simple closed curve)?
Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite ...
- 2,619
3
votes
0
answers
40
views
Non-existence of local generators for Sobolev tangent subundles
Let $U\subset\mathbb R^n$ be a bounded open set, a rank $r\le n$ measurable tangent subbundle $\mathcal V$ on $U$ is a map to the Grassmannian $\mathcal V:U\to Gr(r,\mathbb R^n)$ which is only defined ...
- 938