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We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the…
In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that…
Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations:…
We study the normal state of a 3-$d$ homogeneous dipolar Fermi gas beyond the Hartree-Fock approximation. The correlation energy is found of the same order as the Fock energy, unusually strong for a Fermi-liquid system. As a result, the…
We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i)…
We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of…
In the context of the absolute parallelism formulation of General Relativity, and because of the fact that the scalar curvature can be written in purely torsional terms, it was known for a long time that a surface term based solely on the…
Curvature and helicity topological bounds for the magnetic energy of the streamlines magnetic structures of a kinematic dynamo flow are computed. The existence of the filament dynamos are determined by solving the magnetohydrodynamic…
This is a companion paper to arXiv:1207.3529 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three. In this article we focus on the surface case. A salient feature here…
It is well-known that the Willmore flow of closed spherical immersions exists globally in time and converges if the initial datum has Willmore energy below $8\pi$ - exactly the Li-Yau energy threshold below which all closed immersions are…
We evaluate analytically some ground state properties of two-dimensional harmonically confined Fermi vapors with isotropy and for an arbitrary number of closed shells. We first derive a differential form of the virial theorem and an…
We provide sharp sufficient criteria for an integral $2$-varifold to be induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for $2$-varifolds with critical…
We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian…
We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary…
In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given $3$-dimensional Riemannian…
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive analytically a differential equation connecting these densities, which so far have been…
Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function…
We introduce a parametric framework for the study of Willmore gradient flows which enables to consider a general class of weak, energy-level solutions and opens the possibility to study energy quantization and finite-time singularities. We…
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…
We introduce a fourth-order Willmore-type problem for closed four-dimensional submanifolds immersed in $\mathbb{R}^n$ and establish a connected sum energy reduction for the general fourth-order Willmore energy, analogous to the seminal…