arXiv research

New gravitational instantons and non-holomorphic minimal spheres discovered.

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

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

Study transverse $J$-holomorphic curves linking nearly Kähler $\mathbb{CP}^3$ to minimal surfaces.

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 Fubini-Study metric minimizes a volume-normalized holomorphic systole in $\mathbb{C}P^n$.

This work extends holomorphic surface representations to isotropic space.

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which ar…

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

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

A formula connects discrete harmonic surfaces to holomorphic functions.

Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We…

The paper proves conditions for minimal surfaces to be holomorphic and stable.

Develops explicit formulas for minimal immersions in 5D space.

Minimal surfaces can be mapped to 3D with bounded images.

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic c…

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${…

Constructs continuous families of minimal surfaces and holomorphic immersions.

Every nonflat conformal minimal surface is homotopic to a proper one.

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal sur…

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

Minimal Kaehler submanifolds up to codimension four are studied.

Unified representation for minimal and constant mean curvature surfaces.

New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.

We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil.

Classifies area-minimizing surfaces in R^4 as algebraic.

The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…

A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice ve…

The study examines stable minimal surfaces in higher dimensions and provides bounds on their properties.

Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.

It is shown that if a minimal ruled surface admits a Kähler Yamabe minimizer, then this metric must be generalized Kähler-Einstein and the underlying holomorphic vector bundle of the ruled surface must be quasi-stable.

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…

A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquel…

The paper shows that the Gauss map of minimal surfaces is open and meagre in the space of holomorphic maps.

This paper connects Laguerre minimal surfaces to Weierstrass representations.

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient …

Minimal discs count as knot invariants in hyperbolic 4-space.

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection $(M^4,c,D)$. We show that there is an Eells-Salamon type correspondence between nonvertical $\mathcal{J}$-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces.…

We study general properties of holomorphic isometric embeddings of complex unit balls $\mathbb B^n$ into bounded symmetric domains of rank $\ge 2$. In the first part, we study holomorphic isometries from $(\mathbb B^n,kg_{\mathbb B^n})$ to $(Ω,g_Ω)$ with non-minimal isometric constants $k$ for any irreducible bounded s…

In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.

The paper extends a theorem about stable minimal surfaces to higher codimensions.

A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation between area minimizers and complex curves. When the Kaehler surface is a K3 surfa…

We study proper holomorphic maps between bounded symmetric domains $D$ and $Ω$. In particular, when $D$ and $Ω$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper holomorphic map from $D$ to $Ω$ is a totally geodesic holomorphic isometric embedding with r…

This paper studies ruled real hypersurfaces in indefinite complex projective space.

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a c…

In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.