arXiv research

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 $\mathbb R^n$.

The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS…

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

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructio…

Classifies $C_p$-surfaces using equivariant surgery methods.

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.

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

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.

Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.

Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.

Paper calculates eigenvalues of a specific triangle on a sphere.

The paper creates symmetrical discrete minimal nets using Schwarz reflection.

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…

Develops Brunn-Minkowski theory in hyperbolic space using hyperbolic p-sum.

Stochastic Schwarz lemma on Kähler manifolds via couplings.

Survey explores interactions between four conformal dynamics branches.

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.

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…

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…

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…

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

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.

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved Riemann surfaces M and N whose sectional curvatures $σ_M$ and $σ_N$ satisfy $infσ_M …

We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least arcsinh(2/\sqrt{3}). This lower bound is sharp independently of the topology of S. Th…

Sharp estimate on harmonic maps at conformal points in balls.

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bis…