arXiv research

Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.

Let $(X,ω)$ be a compact Kähler manifold. We introduce and study the largest set $DMA(X,ω)$ of $ω$-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all…

The paper proves a function extension on Kähler manifolds.

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…

Geodesics in non-Archimedean metrics are continuous.

We study the regularizing properties of complex Monge-Ampère flows on a Kähler manifold $(X,ω)$ when the initial data are $ω$-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solutio…

Study on Lee classes of complex surfaces, proving connectedness and bounds.

Lecture notes on using non-Archimedean geometry for complex variety degenerations.

Study proves long-term solutions to a specific equation on hyperKähler manifolds.

The paper proves boundedness of envelopes in complex manifolds.

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, param…

By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $D\subset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical r…

Study geodesic distances and convexity in contact sets.

Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold $(X,ω)$. We show that the complex Monge-Ampère operator $(ω+ dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,ω)$ of $ω$-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class ${\mathcal E}(X,ω)$ is the la…

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given co…

Defines plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.

Let $X$ be a compact Kähler manifold. Given a big cohomology class $\{θ\}$, there is a natural equivalence relation on the space of $θ$-psh functions giving rise to $\mathcal S(X,θ)$, the space of singularity types of potentials. We introduce a natural pseudometric $d_{\mathcal {S}}$ on $\mathcal S(X,θ)$ that is non-de…

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…

Extends complex manifold structures to line bundles, revealing new projective manifolds.

We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continu…

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass a…

Generalizes mean-value inequality to orbifold setting.

Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.

Suppose $(X,ω)$ is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials $\mathcal H$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $(\mathcal H,d)$. We prove that the metric completion of $(\mathcal H,d)$ can be identified with …

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

Let $G/H$ be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold $Γ\big\backslash G/H$ is by definition a quotient of $G/H$ by a discrete uniform subgroup $Γ\leq G$. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…

Given $(X,ω)$ compact Kähler manifold and $ψ\in\mathcal{M}^{+}\subset PSH(X,ω)$ a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that $\int_{X}(ω+dd^{c}ψ)^{n}>0$, we prove that the $ψ-$relative finite energy class $\mathcal{E}^{1}(X,ω,ψ)$ becomes a complete metric space…

Study of non-archimedean μ-entropy and its connection to K-stability.

New neural network models for complex functional data analysis.

Distance function to a finite set is a topological Morse function.

Introduces new weighted floating functions and affine surface areas.

Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.

Neural networks can approximate functionals on RKHS with error bounds.

FFBO optimizes functions as inputs and outputs, improving on existing BO methods.

Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…

The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.

Optimally estimates a functional using nuisance function tuning and sample splitting.

The paper extends mixability theory to function-valued forecasts, proving various loss functions are mixable.

The paper proves isoparametric functions on Finsler space forms under specific conditions.

Paper introduces a nonparametric functional graphical model for random functions.

Robustifies elicitable functionals to handle small distribution misspecifications.

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

Study biharmonic functions on vector bundles with spherical symmetry.

Two new methods improve forecasting of functional time series data.

This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.