arXiv research

Survey explores interactions between four conformal dynamics branches.

The paper creates symmetrical discrete minimal nets using Schwarz reflection.

We give a proof of the classical Schwarz reflection principle for Jenkins-Serrin type minimal surfaces in the homogeneous three manifolds $E(κ, τ)$ for $κ\leqslant 0$ and $τ\geqslant 0$. In our previous paper we proved a reflection principle in Riemannian manifolds. The statements and techniques in the two papers are d…

Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz refl…

Study rationality of meromorphic functions between real algebraic sets in the plane.

The goal of this work is to establish a proof of the Gromov convergence in Hoelder spaces for curves with a totally real boundary condition following the original geometric idea of Gromov. We use a local reflection principle in neighbourhoods of the totally real submanifold as developed by Ivashkovich and Shevchishin a…

New Schwarz Lemma for Bergman metrics in bounded domains.

Explains the Schwarz lemma in lecture notes.

The paper extends Schwarz's lemma to RC-positivity and complex manifolds.

Unified Schwarz lemma in Kähler and Hermitian geometry.

A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can ar…

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

Schwarz lemma extended to equality cases and curvature on manifolds.

Paper introduces capillary Schwarz symmetrization in half-space.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

Higher dimensional generalizations of Schwarz's $P$-surface, Schwarz's $D$-surface and Scherk's second surface are constructed as complete embedded periodic minimal hy- persurfaces in $\mathbb R^n$.

Overview of dynamics in algebraic correspondences and their connections.

Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.

After establishing the uniqueness of the continuation of local Cauchy data for harmonic maps between two Riemannian manifolds M and N, we prove (i) a reflection principle for a smooth minimal submanifold Y of a Riemannian manifold M that contains a reflective submanifold of M as a hypersurface and (ii) the reflection p…

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

Paper calculates eigenvalues of a specific triangle on a sphere.

The paper generalizes the Cauchy-Schwarz-Bunyakovsky inequality and applies it to elasticity problems.

Curve shortening problem solved via Schwarz function.

It has been shown that the Alvarez-Gaum$\mathrm{\acute{e}}$-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…

Stochastic Schwarz lemma on Kähler manifolds via couplings.

Paper generalizes Schwarz lemma for polydisc mappings with specific metrics.

Describes the space of spherical triangles on a smooth 3-manifold.

The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete Riemannian manifolds. Jeffres obtained Schwarz lemma for volume forms of conical Kähler metrics, based on a barrier function an…

Simplified Milnor-Schwarz lemma for geometric group theory.

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

The paper proves a Schwarz lemma for mappings between specific types of manifolds.

We prove estimates interpolating the Schwarz Lemmata of Royden-Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative $\Ric_\ell$ or posit…

In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. T…

Study characterizes cohomology and homotopy types for M-theory extensions.

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Le…

Introduces Cauchy-Schwarz divergence for domain adaptation.

Through the Schwarz lemma, we provide a new point of view on three well-known results of the geometry of hyperbolic surfaces. The first result deal with the length of closed geodesics on hyperbolic surfaces with boundary (Thurston, Parlier, Papadopoulos-Théret). The two others give sharp lower bounds on two metric inva…

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we stu…

We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Sc…

We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…

The paper defines and studies hyperbolicity in calibrated manifolds and derives Schwarz lemmas.

We establish in this note some Cauchy-Schwarz-type inequalities on compact Kähler manifolds, which generalize the classical Khovanskii-Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed Hodge-Riemann bilinear relations due to Dinh and Nguy$\hat{\text{e}}$n. A proportionality p…

The space of polygons up to similarity is studied using the Schwarz-Christoffel formula.