The Schwarz Lemma, a cornerstone in complex analysis, offers crucial insights into the behavior of holomorphic functions on the unit disk. Beyond its classical formulation, this result has significant geometric implications, particularly in the context of minimal surfaces, which are surfaces with zero mean curvature. These surfaces arise in both differential geometry and physics, with applications ranging from soap films to general relativity. The conformality of harmonic maps, especially those that preserve area, is fundamental in understanding minimal surfaces, and the Schwarz Lemma provides key constraints on these maps. It helps in establishing curvature bounds and offers tools for analyzing the behavior of minimal surfaces near their boundaries. In this presentation, we will explore two key results that extend the reach of the Schwarz Lemma in the theory of minimal surfaces: (1) Extension of the Schwarz Lemma for Conformal Parameterization of Minimal Surfaces: We will introduce a generalized version of the Schwarz Lemma that applies to conformal mappings associated with minimal surfaces, offering a new perspective on their parameterization. (2) Solution to the Gaussian Curvature Conjecture for Minimal Graphs: We will present a solution to the long-standing conjecture related to the Gaussian curvature of minimal graphs, shedding light on their intrinsic geometry. These results showcase the deep interplay between complex analysis and geometric properties of minimal surfaces, revealing new avenues for research and applications.

The Schwarz Lemma, a cornerstone in complex analysis, offers crucial insights into the behavior of holomorphic functions on the unit disk. Beyond its classical formulation, this result has significant geometric implications, particularly in the context of minimal surfaces, which are surfaces with zero mean curvature. These surfaces arise in both differential geometry and physics, with applications ranging from soap films to general relativity. The conformality of harmonic maps, especially those that preserve area, is fundamental in understanding minimal surfaces, and the Schwarz Lemma provides key constraints on these maps. It helps in establishing curvature bounds and offers tools for analyzing the behavior of minimal surfaces near their boundaries. In this presentation, we will explore two key results that extend the reach of the Schwarz Lemma in the theory of minimal surfaces: (1) Extension of the Schwarz Lemma for Conformal Parameterization of Minimal Surfaces: We will introduce a generalized version of the Schwarz Lemma that applies to conformal mappings associated with minimal surfaces, offering a new perspective on their parameterization. (2) Solution to the Gaussian Curvature Conjecture for Minimal Graphs: We will present a solution to the long-standing conjecture related to the Gaussian curvature of minimal graphs, shedding light on their intrinsic geometry. These results showcase the deep interplay between complex analysis and geometric properties of minimal surfaces, revealing new avenues for research and applications.

We consider the Gaussian curvature conjecture of a minimal graph \(S\) over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk's type minimal surfaces over a quadrilateral inscribed in the unit disk containing the origin inside. As an application we improve so far the obtained upper estimates of  Gaussian curvature at the point above the center. Further we obtain an optimal estimate of the Gaussian curvature at the point \(\mathbf{w}\) over the center of the disk, provided \(\mathbf{w}\) satisfies certain ''symmetric'' conditions. The result extends a classical result of Finn and Osserman in 1964. In order to do so, we construct a certain family \(S^t\), \(t\in[t_\circ, \pi/2]\) of Scherk's type minimal graphs over the isosceles trapezoid inscribed in the unit disk. Then we compare the Gaussian curvature of the graph \(S\) with that of \(S^t\) at the point \(\mathbf{w}\) over the center of the disk.