arXiv research

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary map…

The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.

Maps with many singularities found in complex space.

Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\displaystyle f^*h - \frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the …

Proves continuity and singular set dimension for 2D maps with Q values.

For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $ΣM$, and any compact Riemannian manifold $N$, we show an $ε$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak converg…

Stable harmonic maps into certain Lie groups have singularities with specific codimensions.

Formula derived for blow-up of quaternionic maps on Hyperkähler manifolds.

New energy identity found for biharmonic maps into spheres.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $Π$ is Fredholm by showing that the stationary vacuum equations (combined with p…

Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.

Algorithm adapts to non-stationary rewards without prior knowledge.

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all…

Bayesian method improves SOM training for dynamic data.

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determin…

The paper proves an energy identity for harmonic maps near singularities.

In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bo…

Study on financial systems using perturbed unimodal maps with heteroscedastic noise.

Detect changes in noisy dynamical systems using empirical approximations and finite-sample bounds.

Strong stability of ergodic iterations proven without ergodic driving sequence.

Unified review of methods for inferring non-stationary process parameters.

The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\smΣ\to\H^{k+1}_\C$ into the $(k+1)$-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed sing…

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of varia…

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $\int_M|du_p|^p=O(\frac{1}{2-p}).$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the sup…

We provide a geometric framework for the construction of non-vacuum black holes whose metrics are stationary and axisymmetric. Under suitable assumptions we show that the Einstein equations reduce to an Einstein-harmonic map type system and analyze the compatibility of the resulting equations. This framework will be fu…

New theory for area of Legendrian surfaces, proving smoothness and variational results.

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometr…

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

We present a novel approach for fully non-stationary Gaussian process regression (GPR), where all three key parameters -- noise variance, signal variance and lengthscale -- can be simultaneously input-dependent. We develop gradient-based inference methods to learn the unknown function and the non-stationary model param…

Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class $W^{1,p}$, $1 < p < \infty$. If $f$ almost minimizes its $p$ Dirichlet energy then $f$ is Hölder continuous. If $p=2$ and $f$ is squeeze and squash stationary then $f$ is in VMO.

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, howeve…

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Optimizes biharmonic map regularity using stratification methods.

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Li…

In this paper we generalize harmonic maps and morphisms to the \emph{degenerate semi-Riemannian category}, in the case when the manifolds $M$ and $N$ are \emph{stationary} and the map $φ:M\to N$ is \emph{radical-preserving}. We characterize geometrically the notion of \emph{(generalized) horizontal (weak) conformality}…

Study proves existence of non-trivial harmonic map flows to hemispheres.

Recovering matrix valued potentials from wave equation data on stationary spacetimes.

Building models, or maps, of robot environments is a highly active research area; however, most existing techniques construct unstructured maps and assume static environments. In this paper, we present an algorithm for learning object models of non-stationary objects found in office-type environments. Our algorithm exp…

Random walks on mapping class groups identified with geodesic laminations.

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic m…

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

Study stationary measures and orbit closures for non-abelian actions on surfaces.

We characterize stationary solutions to McKean-Vlasov equations on the circle.

The paper calculates Morse indices and nullities for embedded networks on spheres.

PROPO tackles non-stationary MDPs with efficient policy optimization.

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets o…

The Cannon-Thurston map's measures become singular with respect to sphere measures.

The state space (SS) representation of Gaussian processes (GP) has recently gained a lot of interest. The main reason is that it allows to compute GPs based inferences in O(n), where $n$ is the number of observations. This implementation makes GPs suitable for Big Data. For this reason, it is important to provide a SS …