arXiv research

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that…

Paper proves Toponogov's theorem in Alexandrov geometry.

New proof of Gaffney's inequality for differential forms on manifolds with boundary.

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.

New proof for global rigidity of vertex scaling on polyhedral surfaces.

Minimal submanifolds are found as energy concentration sets in variational problems.

New guarantees for black-box variational inference methods.

We give a new and detailed proof of the variation formulas for the equivariant Ray-Singer metric, which are originally due to J.M. Bismut and W. Zhang.

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main d…

New mass inequalities and proofs for causal variational principles.

This short note contains an explicit proof of the Jacobi identity for variational Schouten bracket in $Z_2$-graded commutative setup. For the reasoning to be rigorous, it refers to the product bundle geometry of iterated variations (see arXiv:1312.1262 [math-ph]); no ad hoc regularizations occur anywhere in this theory…

In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

This is a sequel of our paper [arXiv:1809.08425] on the Quot-scheme limit and variational properties of Donaldson's functional, which established its coercivity for slope stable holomorphic vector bundles over smooth projective varieties. Assuming that the coercivity is uniform in a certain sense, we provide a new proo…

A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems.

Proofs for spectral and geometric properties of hyperbolic surfaces.

The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulatio…

We reinterpreting the variational inference in a new perspective. Via this way, we can easily prove that EM algorithm, VAE, GAN, AAE, ALI(BiGAN) are all special cases of variational inference. The proof also reveals the loss of standard GAN is incomplete and it explains why we need to train GAN cautiously. From that, w…

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly …

A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all spectral curves corresponding to periodic solutions of the Davey-Stewartson equation…

In this paper, using the Greiner's approach to heat kernel asymptotics, we give new proofs of the equivariant Gauss-Bonnet-Chern formula and the variation formulas for the equivariant Ray-Singer metric, which are originally due to J. M. Bismut and W. Zhang.

We study the geometry of jets of submanifolds with special interest in the relationship with the calculus of variations. We give a new proof of the fact that higher order jets of submanifolds are affine bundles; as a by-product we obtain a new expression for the associated vector bundles. We use Green-Vinogradov formul…

In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's $ν$-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly …

In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and …

Study optimizes perimeter in convex domains with anisotropic constraints.

We prove that an analog of the exterior differential acts on the space of arbitrary Lagrangians of multidimensional paths on any manifold or supermanifold, thus making this space into a cochain complex. An analog of the Stokes' formula holds. The construction and the proofs are purely geometrical, in terms of the varia…

The geometric Lagrangian theory (of arbitrary order) is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc. In this paper we give new and much simpler proofs for the whole theory using Fock…

New, shorter proofs for varifolds and flows with improved decay of flatness.

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' appro…

Given a special Kahler manifold M, we give a new, direct proof of the relationship between the quaternionic structure on its cotangent bundle and the variation of Hodge structures on the complexification of TM.

We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_…

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existen…

New proof given for a functional's minimum condition.

We provide a constructive, variational proof of Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corres…

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian the…

Proof of Graustein's theorem in different geometries.

Discrete conformal maps on surfaces with vertex decorations are studied.

We investigate the optimization of two probabilistic generative models with binary latent variables using a novel variational EM approach. The approach distinguishes itself from previous variational approaches by using latent states as variational parameters. Here we use efficient and general purpose sampling procedure…

We propose a molecular generative model based on the conditional variational autoencoder for de novo molecular design. It is specialized to control multiple molecular properties simultaneously by imposing them on a latent space. As a proof of concept, we demonstrate that it can be used to generate drug-like molecules w…

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(λ_i)_{i=1}^\infty$ be the variational spectrum of $Δ_p$ on a closed Riemannian manifold $(X,g)$ and let $N(λ) = \#\{i:\, λ_i < λ\}$ be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorna…

We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component links in $S^3$ up to $Δ$-link homotopy. We also prove its version for string li…

In this paper we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the epsilon-Dirichlet problem obtained as a variational problem for the SU(2)-epsilon-Yang Mills functional. This is equivalent to proving the existence of multiple solutions to the Dirichlet problem fo…

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) fo…

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations a…

In this survey article we describe the geometry of toric hyperkähler varieties, which are hyperkähler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation of hyperkähler structures of these spaces with many improved results and proofs.

In a family of compact, canonically polarized, complex manifolds equipped with Kähler-Einstein metrics the first variation of the lengths of closed geodesics was previously shown in by the authors in [arXiv:0808.3741v2] to be the geodesic integral of the harmonic Kodaira-Spencer form. We compute the second variation. F…