arXiv subjects

Dominik Inauen

Publications and source records attributed to Dominik Inauen.

14 recordsLinked to original sources

arXiv2026

Full flexibility of the Monge-Amp\`ere system in codimension $d_*-d+1$

We prove that $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Amp\`ere system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every H\"older exponent $\alpha<1$. Our result strengthens the statement in [Lewicka 2022], obtained for $k = 2d_*$ and based on ideas from [K\"allen 1978] in the context of the isometric immersion system. It also generalizes the result of [Inauen-Lewicka 2025], where full flexibility was established in dimension $d=2$ and codimension $k=2$. The same proof scheme further yields local full flexibility of isometric immersions of $d$-dimensional Riemannian metrics into Euclidean space of dimension $d_* + 1$, generalizing the result in [Lewicka 2025] proved for $d=k=2$. By using techniques of [Conti-De Lellis-Szekelyhidi], the result can be extended to compact manifolds, in codimension $(d+1)d_*-d+1$.

math.AP↗

arXiv2026

Flexibility of Codimension One $C^{1,\theta}$ Isometric Immersions

We study the problem of constructing $C^{1,\theta}$ isometric immersions of Riemannian metrics on $n$-dimensional domains into $\mathbb{R}^{n+1}$. While the classical Nash--Kuiper theorem established the flexibility of $C^1$ isometries, subsequent work has extended this to $C^{1,\theta}$ isometries for certain $\theta$, though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by $C^{1,\theta}$ isometric immersions for $\theta< 1/(1+2(n-1))$, improving upon the previously known exponent for $n\geq 3$. The improvement is obtained via a convex integration scheme incorporating a refined iterative integration by parts procedure resting on a detailed structural analysis of error terms and the interaction of multiple frequency scales.

math.AP↗

arXiv2025

Isometric Immersions and Weak Solutions to the Darboux Equation

We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in C^{1,\theta}$ with $\theta>1/2$, and show that the classical correspondence between solutions of the Darboux equation and isometric immersions remains valid in this regime. The key ingredient is an extension of the classical flatness criterion to H\"older continuous metrics, achieved via an analysis of a weak notion of Gaussian curvature.

math.AP↗

arXiv2025

The Monge-Amp\`ere system in dimension two is fully flexible in codimension two

We prove that every $\mathcal{C}^1(\bar\omega)$-regular subsolution of the Monge-Amp\`ere system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with regularity $\mathcal{C}^{1,\alpha}(\bar\omega)$ for any $\alpha<\min\{1, \frac{s+\beta}{2}\}$, where $\mathcal{C}^{s,\beta}$ is the assumed regularity of the system's right hand side. This result suggests the full flexibility of Poznyak's theorem for isometric immersions of $2$d Riemannian manifolds into $\mathbb{R}^4$, and asserts it in the parallel setting of the Monge-Amp\`ere system.

math.AP↗

arXiv2025

A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent

Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any H\"older exponent $\alpha<(1+n^2-n)^{-1}$, we construct a $C^{1, \alpha}$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of $C^{1, \alpha}$ isometric immersions beyond Borisov's exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1/3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors.

math.AP↗

arXiv2025

The Monge-Ampere system in dimension two and codimension three

We revisit the convex integration constructions for the Monge-Amp\`ere system and prove its flexibility in dimension $d=2$ and codimension $k=3$, up to $\mathcal{C}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the obtained H\"older exponent $1-\frac{1}{\sqrt{5}}$ is larger than $1/2$ but it is not contained in the full flexibility up to $\mathcal{C}^{1,1}$ result. Previous various approaches, based on Kuiper's corrugations, always led to the H\"older regularity not exceeding $\mathcal{C}^{1,1/2}$, while constructions based on the Nash spirals (when applicable) led to the regularity $\mathcal{C}^{1,1}$. Combining the two approaches towards an interpolation between their corresponding exponent ranges has been so far an open problem.

math.AP↗

arXiv2023

Stochastic Nash evolution

This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $\alpha \in ]\tfrac{1}{2},1]$, we show that a $C^{1,\alpha}$ embedding $u: M \to \mathbb{R}^q$ is isometric if and only if the intrinsic and extrinsic constructions of Brownian motion on $u(M)\subset \mathbb{R}^q$ yield processes with the same law. The equivalence is first established for smooth embeddings; this is followed by a renormalization procedure for $C^{1,\alpha}$ embeddings. In particular, we also construct extrinsic Brownian motion when $g \in C^2$ and $u$ is a $C^{1,\alpha}$ isometric embedding. This formulation is based on a gedanken experiment that relates the intrinsic and extrinsic constructions of Brownian motion on an embedded manifold to the measurement of geodesic distance by observers in distinct frames of reference. This viewpoint provides a thermodynamic formalism for the isometric embedding problem that is suited to applications in geometric deep learning, stochastic optimization and turbulence.

math.PR↗

arXiv2023

$C^{1,\frac{1}{3}-}$ very weak solutions to the two dimensional Monge-Amp\'ere equation

For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.

math.AP↗

arXiv2022

Dyson Brownian Motion and motion by mean curvature

We construct Dyson Brownian motion for $\beta \in (0,\infty]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $\beta$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.

math.PR↗

arXiv2020

Rigidity and Flexibility of Isometric Extensions

In this paper we consider the rigidity and flexibility of $C^{1, \theta}$ isometric extensions and we show that the H\"older exponent $\theta_0=\frac12$ is critical in the following sense: if $u\in C^{1,\theta}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $\Sigma$ and $\theta> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $\Sigma$. On the other hand, for any $\theta<\frac12$ we can construct $C^{1,\theta}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, \theta}$ isometric embeddings, $\theta<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.

math.AP↗

arXiv2019

$C^{1,α}$ Isometric Embeddings of Polar Caps

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the Hölder space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $α> \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $α< \frac{1}{2}$ we construct $C^{1,α}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.

math.AP↗

arXiv2018

Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of "A selection principle for the sharp quantitative isoperimetric inequality" by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.

math.AP↗

arXiv2017

Fractional Sobolev Regularity for the Brouwer Degree

We prove that if $Ω\subset \mathbb R^n$ is a bounded open set and $nα> {\rm dim}_b (\partial Ω) = d$, then the Brouwer degree deg$(v,Ω,\cdot)$ of any Hölder function $v\in C^{0,α}\left (Ω, \mathbb R^{n}\right)$ belongs to the Sobolev space $W^{β, p} (\mathbb R^n)$ for every $0\leq β< \frac{n}{p} - \frac{d}α$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $β\geq 0$ and $p\geq 1$ with $β> \frac{n}{p} - \frac{n-1}α$ there is a vector field $v\in C^{0, α} (B_1, \mathbb R^n)$ with $\mbox{deg}\, (v, Ω, \cdot)\notin W^{β, p}$, where $B_1 \subset \mathbb R^n$ is the unit ball.

math.CA↗

arXiv2016

A Nash-Kuiper theorem for $C^{1,\frac{1}{5}-δ}$ immersions of surfaces in $3$ dimensions

We prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensional disk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb R^3$ can be uniformly approximated with $C^{1,α}$ isometric immersions for any $α< \frac{1}{5}$. This statement improves previous results by Yu.F. Borisov and of a joint paper of the first and third author with S. Conti.

math.DG↗