All Questions
Tagged with dg.differential-geometry real-analysis
158 questions
2
votes
0
answers
85
views
Are a map with constant singular values and its inverse always conjugate through isometries?
Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...
- 6,741
2
votes
0
answers
269
views
Extending Green's theorem from very special regions to more general regions
Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
- 151
2
votes
0
answers
80
views
Generalized definition of integrable condition on rough complex subbundle
Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...
- 938
2
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0
answers
262
views
Are $C^1$ immersions dense in $C^1$?
Let $M$ be a closed compact manifold.
Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
- 43
2
votes
1
answer
287
views
Regularity of the reparametrization map between curves [closed]
I am looking for a reference for the following kind of results.
Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm.
Let $B$ be a Borel subset of ...
- 980
2
votes
0
answers
194
views
A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 505
2
votes
0
answers
202
views
Universal chord theorem for curves
Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
- 121
2
votes
0
answers
67
views
On two functions with isodirectional gradients
Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$
\begin{equation}
(\...
- 1,461
2
votes
0
answers
184
views
Modify the jump set of $BV$ function
Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
- 679
1
vote
1
answer
261
views
The existence of differential operator of the form $AB=0$
We define $\mathcal A$ is a differential operator of order $n$ with variable coefficients if
$$ \mathcal A:=\sum_{|\alpha|\leq n}A_\alpha (x) D^\alpha $$
where $\alpha$ is an muti-index and $A_\alpha(...
- 679
1
vote
2
answers
235
views
Can we get smooth parition of unity with uniformity?
Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of ...
- 3,625
1
vote
1
answer
342
views
Singular conformally-Euclidean metrics
Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':
$$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{...
- 2,222
1
vote
1
answer
345
views
Topological degree of differentiable map using line integrals?
Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
1
vote
2
answers
83
views
Existence of Smooth path in a Domain through a Sequence of Points
The following question seems intuitively true, but I'm unable to see the proof. While I could prove it when $U=\mathbb{R}^n$, but for other open sets, I do not have a proof. Although I could construct ...
- 895
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
$$\begin{align}
\sigma \colon I \...
- 965
1
vote
1
answer
447
views
A distribution $u$ such that all of its derivatives are of order zero is smooth
I'm reading Demailly's Complex Analytic and Differential Geometry In Section I.2.D.4 he uses the following fact: Suppose $u \in \mathcal{D}'(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a ...
- 1,141
1
vote
1
answer
146
views
Is a locally invertible weak limit of injective maps injective almost everywhere?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
- 6,741
1
vote
1
answer
674
views
Directional gradient on sphere
We consider the following function
$$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x_1,...,x_N)= \sum_{i=1}^N x_i.$$
This function can be written in Cartesian coordinates as $f(x)=...
- 536
1
vote
1
answer
1k
views
Approximation of a continuous function by a smooth one on an open set
I'm interested in the following kind of theorems :
Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
- 193
1
vote
1
answer
980
views
Extension of a smooth function from a convex set
Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear ...
- 1,424
1
vote
1
answer
191
views
Sequence of smooth maps converging to the identity [closed]
Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
- 27
1
vote
1
answer
160
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
1
vote
1
answer
163
views
Argument for differentiability of a certain quotient of smooth functions
I have what is in essence a basic analysis question.
To make working out a certain example a bit easier I found that I need to find existence of a function $f\in C^\infty(\mathbb{R})$ with the ...
- 397
1
vote
1
answer
161
views
Continuity of image of resolvent operator with respect to resolvent parameter
Suppose $D$ is a first-order differential operator on a manifold $M$ and that the inverse $(D+t)^{-1}:H^0(M)\rightarrow H^1(M)$ exists for all $t > 0$, where $H^i(M)$ is the $i^\text{th}$ Sobolev ...
- 1,903
1
vote
1
answer
307
views
Ordinal of injectivity for a smooth regular curve with a finite arc-length
Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve ...
- 43
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
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
1
vote
0
answers
155
views
Does there always exist a regular curve connecting two points in an open connected subset of $\mathbb{R}^n$? [closed]
As the title says, given $A\subseteq \mathbb{R}^n$ open and connected and $x, y\in A$, I am looking for a continuous curve $\gamma:[0, 1]\rightarrow A$ which is differentiable in $(0,1)$ with $\gamma'(...
- 21
1
vote
0
answers
131
views
Integral flow that can commute to Laplacian operator
Firstly, considering the vector field in $ \mathbb{R}^3 $, $ X=x_2e_1-x_1e_2 $, we can see that
$$
\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
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
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
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
1
vote
0
answers
88
views
Density of $C^k$-functions with Lipschitz partial derivatives
Let $N$ and $M$ be complete Riemannian manifolds, of respective dimension $n$ and $m$ with $n,m\geq 1$. Let $C^{k,1}_b(N,M)$ be set of all bounded continuous functions $f:N\rightarrow M$ for which ...
- 5,405
1
vote
0
answers
145
views
Integrability conditions imply existence of potential
I'm looking for a proof of the following well-known theorem:
If $f$ is a continuously differentiable vector field in a simply connected region $G\subset \mathbb{R}^n$ which satisfies the ...
- 2,160
1
vote
0
answers
55
views
Projection of a real analytic manifold onto subspace is union of real analytic submanifolds
Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
- 19
1
vote
0
answers
83
views
Gradient descent in $U(n)^r$
I have a function $f:U(n)^r\rightarrow \mathbb{R}$ which I would like to minimize. Here, $U(n)$ is the set of unitary matrices, and $r$ should be considered to be much bigger than $n$. For instance, $...
- 111
1
vote
0
answers
62
views
Regularity of a shrunken domain
I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.
Let $\Omega\subset\Bbb R^d$ be an open bounded (...
- 1,101
1
vote
0
answers
242
views
Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]
[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
- 133
1
vote
0
answers
96
views
System of Poisson equations
Let $(M,g)$ be a closed (compact and without boundary) and oriented Riemannian manifold and let us consider the Poisson equation for a smooth function $\varphi$:
$\Delta \phi = f$,
where $f$ is a ...
- 2,816
1
vote
0
answers
112
views
Question regarding the image of a polynomial map containing a small box
I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously.
Let $\delta, \varepsilon > 0$.
Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
- 3,625
1
vote
0
answers
69
views
The jump set of $SBV$ function over a hyper surface
Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$.
Now, given a positive ...
- 679
1
vote
0
answers
196
views
Extension of diffeomorphisms preserving bilateral bounds of the derivatives
Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
- 31
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
0
votes
1
answer
115
views
Average over spheres finite
Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$
I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has ...
- 124
0
votes
1
answer
1k
views
Global Implicit Function Theorem
Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ such that
$F(x,\theta(z,x))=z$ for ...
- 13
0
votes
2
answers
132
views
Dirichlet problem for capillary equation over convex domain
Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary.
Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function.
Let $L$ be a quasilinear elliptic ...
- 823
0
votes
1
answer
87
views
Differentiablity of certain composite function
Let $I_1$ and $I_2$ be two closed bounded intervals.
Suppose $W(x,y)$ is a smooth function whose support is contained inside $I_1 \times I_2$.
Suppose I have $\Phi= (\Phi_1(x,y), \Phi_2(x,y)) : \...
- 3,625
0
votes
1
answer
94
views
A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?
Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...
- 109
0
votes
1
answer
182
views
Surjectivity of "nice maps" from local properties
What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...
- 1,256
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
0
votes
1
answer
71
views
Regular curve given implicitly
Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application.
What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
- 33