arXiv research

Paper proves unique energy-minimizing curves in constrained spaces.

The paper classifies energy-minimizing sets in specific domains.

Study energy-minimizing maps in projective spaces, proving sharp bounds.

We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.

We show $C^{1,α}$-regularity for energy minimizing maps from a 2-dimensional Riemannian manifold into a Finsler space $(\R^n, F)$ with a Finsler structure $F(u,X)$.

New theorem on 3-manifolds with curvature and convex boundary.

In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's th…

New proof of harmonic map uniqueness with analytic targets.

Study on biharmonic almost complex structures on compact manifolds.

Rectifies singular set of harmonic maps into complex.

The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete ou…

We prove that energy minimizing Yang-Mills connections on a compact $G_{2}$-manifold has holonomy equal to $G_{2}$ are $G_{2}$-instantons, subject to an extra condition on the curvature. Furthermore, we show that energy minimizing connections on a compact Calabi-Yau $3$-fold has holonomy equal to $SU(3)$ subject to a s…

In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of riemannian polhyedra without focal points. Among other things, we show that on the relevant polyhedr…

We study optimal double helices with straight axes (or the fattest tubes around them) computationally using three kinds of functionals; ideal ones using ropelength, best volume packing ones, and energy minimizers using two one-parameter families of interaction energies between two strands of types $r^{-α}$ and $\frac1r…

Introduces Causal Energy Minimization to understand Transformer layers.

New taxonomy and improved solvers for discrete energy minimization.

We present a new proximal bundle method for Maximum-A-Posteriori (MAP) inference in structured energy minimization problems. The method optimizes a Lagrangean relaxation of the original energy minimization problem using a multi plane block-coordinate Frank-Wolfe method that takes advantage of the specific structure of …

There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic co…

Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.

The paper studies harmonic graphs in the Heisenberg group and their properties.

This work develops machine learning for micromagnetic energy minimization.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

Study axisymmetric surfaces in Euclidean space for energy minimization.

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show t…

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…

Optimizes energy of mappings from complex projective spaces.

Improved optimal regularity for harmonic almost complex structures.

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - chara…

The article analyzes the stability of a curve shortening flow for planar networks.

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

Let $N=(Ω,σ)$ and $M=(Ω^*,ρ)$ be doubly connected Riemann surfaces and assume that $ρ$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms between $Ω$ and $Ω^*$ that minimize the Dirichlet energy. In the class of all homeomorphisms $f \col…

Unique geodesics selected by energy minimization in Teichmüller space.

Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.

A new deep learning method for option pricing in rough volatility models.

In this thesis I explore challenging discrete energy minimization problems that arise mainly in the context of computer vision tasks. This work motivates the use of such "hard-to-optimize" non-submodular functionals, and proposes methods and algorithms to cope with the NP-hardness of their optimization. Consequently, t…

Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.

We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons.

Derives equilibrium law for Plateau borders in wet soap films and foams.

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos-Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theor…

We show that for every quasi-isometric map from a Hadamard manifold of pinched negative curvature to a locally compact, Gromov hyperbolic, ${\rm CAT}(0)$-space there exists an energy minimizing harmonic map at finite distance. This harmonic map is moreover Lipschitz. This generalizes a recent result of Benoist-Hulin.

The study finds minimal distortion embeddings of surfaces into small domains.

Lipschitz mappings found between Riemann surfaces with specific properties.

We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit under the action of volume preserving diffeomorphisms. This minimization property for Beltrami fields is relevant because of its connections with the phenomenon of magn…

Study on harmonic maps in special geometric spaces.

Recently, supervised hashing methods have attracted much attention since they can optimize retrieval speed and storage cost while preserving semantic information. Because hashing codes learning is NP-hard, many methods resort to some form of relaxation technique. But the performance of these methods can easily deterior…

We study the notion of $p$-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every $p$-quasihomotopy class of Newtonian maps contains a minimizer of the $p$-e…

New energy model avoids self-intersections in curve optimization.