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5 votes
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Continuity of Hausdorff measure on level sets

Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that: $\bullet$ $\phi^{-1}(0)\neq\emptyset$; $\bullet$ $\nabla\phi(x)\...
Bogdan's user avatar
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6 votes
0 answers
171 views

The distributional gradient of the closest isometry to the differential of a smooth map

The setting-a "linear algebra" fact: Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
Asaf Shachar's user avatar
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5 votes
0 answers
416 views

Extending Gromov's inequality

In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound $$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol} \frac{\stsys_2^n}{...
Mikhail Katz's user avatar
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5 votes
0 answers
238 views

Is polar decomposition of a smooth map Sobolev?

Motivation: Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
Asaf Shachar's user avatar
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2 votes
0 answers
354 views

Continuity of surface integrals on level sets

Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
Bogdan's user avatar
  • 1,759
1 vote
1 answer
201 views

Does weak continuity of Jacobians hold for non nondegenerate maps?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \...
Asaf Shachar's user avatar
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