arXiv research

Study on polynomial growth functions and forms on gradient Ricci solitons.

Classifies Kähler metrics with constant holomorphic curvature.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

Extends holomorphic functions on complex manifolds to larger spaces.

It is well-known that non-constant holomorphic functions do not exist on a compact complex manifold. This statement is false for a supermanifold with a compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a compact homogeneous complex supermanif…

Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.

Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct exampl…

A formula connects discrete harmonic surfaces to holomorphic functions.

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $μ$ on these groups are studied. In particular, we establish a unitary equivalence between…

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex…

The paper finds many negatively curved Kähler metrics on complex manifolds.

The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.

This paper provides an explicit form for symmetric differentials and their corresponding holomorphic functions.

We gave an alternative short proof on the finite generation of holomorphic functions with polynomial growth on Riemann surfaces with nonnegative curvature. The first proof was due to Li and Tam.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…

The holomorphic torsion of a hermitian locally symmetric space is expressed as a special value of a geometric zeta function.

The article studies critical points of a new energy functional in higher dimensions.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

We present a holomorphic representation of the Jacobi algebra $\mathfrak{h}_n\rtimes \mathfrak{sp}(n,\R)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}^n\times \mathcal{D}_n$. We construct the Hilbert space of holomorphic functions on which these differential operators a…

Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.

A representation of the Jacobi algebra $\mathfrak{h}_1\rtimes \mathfrak{su}(1,1)$ by first order differential operators with polynomial coefficients on the manifold $\mathbb{C}\times \mathcal{D}_1$ is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with …

The conformal invariance and universality results of Chelkak-Smirnov on the two-dimensional Ising model hold for isoradial planar graphs with critical weights. Motivated by the problem of extending these results to a wider class of graphs, we define a generalized notion of s-holomorphicity for functions on arbitrary we…

Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$_d(M^n)$the space of holomorphic functions of polynomial growth of degree at most$d$on$M^n$. In this paper we prove that$dim_{\mathbb{C}}{\mathcal{O}}_d(…

The paper calculates a formula for knot complements using holomorphic curves.

The paper studies special surfaces with a new type of support function.

The holomorphic torsion of a compact locally symmetric manifold is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.

The paper introduces new functionals and equations for complex vector bundles.

We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete Kähler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of cur…

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

We observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe …

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Koda…

The paper proves compactness for holomorphic curves with boundary on nearby Lagrangians.

This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting part…

Holomorphic functions from knot complements link to quantum modular forms.

This work extends holomorphic surface representations to isotropic space.

In this paper, the theory of functions of one complex variable is explored to study linearly full unramified holomorphic two-spheres with constant curvature in $G(2,n)$ satisfying that the generated harmonic sequence degenerates at position $2$. Firstly, we determine the value distribution of the curvature and give the…

Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitab…