arXiv research

Flat isometric immersions in 3D are developable if they are $C^{1,2/3}$ regular.

Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is ob…

Proves harmonic coordinates for weak immersions in even dimensions.

Study shows some contractible complexes can't have certain immersions.

We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling scenarii. As a consequence, out of the twelve families of non-planar minimal surface…

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…

We will study the blowup behavior of a surface sequence immersed in $R^2$ with bounded Willmore functional and fixed genus.

The paper discusses fractional Sobolev immersions of flat domains into 3D space.

Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.

Study shows weak homotopy equivalences for complete minimal surfaces.

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a sub…

Survey on recent developments in isometric immersions using PDE techniques.

Study shows submanifolds can't be immersed in certain spaces.

Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional cur…

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to Hélein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in $\mathbb{R}^{n}$ w…

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…

Study of immersions with Willmore energy leading to spherical and catenoid bubbles.

The study proves a strong parametric h-principle for minimal surfaces.

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (…

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…

A new parametric method studies Willmore flows and energy quantization.

The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces i…

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We…

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (includin…

Doodles link to commutator identities in a 2-sphere.

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential…

The paper classifies special hypersurfaces in space forms.

Smooth low-regular connections lead to smooth immersions with controlled regularity.

We study immersed surfaces in $\mathbb{R}^3$ which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting fre…

Critical points of scale-invariant curvature energies in 4D are analytic.

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. In this paper, we prove the following much stronger result in this direction: for any $n\geq …

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a …

In the present work we study the behavior of sequences of smooth global isothermic immersions of a given closed surface and having a uniformly bounded total curvature. We prove that, if the conformal class of this sequence is bounded in the Moduli space of the surface, it weakly converges in W^{2,2} away from finitely …

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…

In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, R…

We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in $\mathbb{R}^3$ of total curvature greater than $-12π$, only three of them …

We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the…

Study on membranes under confinement, proving existence and regularity of minimizers.

We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes th…

This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of man…

The study proves conditions for constant curvature submanifolds in space forms.

Extending isometric immersions with low regularity, especially supercritical.