Review of geometry and analysis

  title={Review of geometry and analysis},
  author={Shing-Tung Yau},
  journal={Asian Journal of Mathematics},
  • S. Yau
  • Published 2000
  • Mathematics
  • Asian Journal of Mathematics
In this article, we shall discuss what the author considers to be important in geometry and related subjects. Since the time of the Greek mathematicians, geometry has always been in the center of science. Scientists cannot resist explaining natural phenomena in terms of the language of geometry. Indeed, it is reasonable to consider geometric objects as parts of nature. Practically all elegant theorems in geometry have found applications in classical or modern physics. In order to understand the… 
A survey on classical minimal surface theory
Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the
Geometry and Topology
This chapter presents a collection of theorems in geometry and topology, proved in the twenty-first, which are at the same time great and easy to understand. The chapter is written for undergraduate
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
The Calabi-Yau conjectures for embedded surfaces
In this talk I will discuss the proof of the Calabi-Yau conjectures for embedded surfaces. This is joint work with Bill Minicozzi, [CM9]. The Calabi-Yau conjectures about surfaces date back to the
Geometric Variational Problems for Mean Curvature
This thesis investigates variational problems related to the concept of mean curvature on submanifolds. Our primary focus is on the area functional, whose critical points are the minimal submanifolds
Minimal surfaces in Euclidean spaces by way of complex analysis
This is an extended version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications of
Communications in Mathematical Physics Isometric Immersions and Compensated Compactness
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3. This problem can
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These
Holomorphic Legendrian curves in ℂℙ3 and superminimal surfaces in 𝕊4
We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the space
We collect dozens of well-known and not so well-known fundamental unsolved problems involving low dimensional submanifolds of Euclidean space. The list includes selections from differential geometry,


Special Lagrangian Fibrations II: Geometry
We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special
Special Lagrangian Fibrations I: Topology
In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y are a mirror pair of such manifolds, then X
On the Existence of a Closed Convex Surface Realizing a Given Riemannian Metric.
  • H. Lewy
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1938
1 This problem corresponds to Douglas' generalization of Plateau's original problem, cf. the papers by Douglas, Journ. Math. Phys., 15, 55 ff (Feb., 1936) and 106 ff (June, 1936) and Courant, these
Conformal deformation of a Riemannian metric to constant scalar curvature
A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This
Integration over the u-plane in Donaldson theory
We analyze the u-plane contribution to Donaldson invariants of a fourmanifold X. For b2 (X) > 1, this contribution vanishes, but for b + 2 = 1, the Donaldson invariants must be written as the sum of
Zero-loop open strings in the cotangent bundle and Morse homotopy
0. Introduction. Many important works in symplectic geometry and topology are regarded as the symplectization or the quantization of the corresponding results in ordinary geometry and topology. One
Compact Riemannian 7-manifolds with holonomy $G\sb 2$. II
This is the second of two papers about metrics of holonomy G2 on compact 7manifolds. In our first paper [15] we established the existence of a family of metrics of holonomy G2 on a single, compact,
Asymptotic-behavior for singularities of the mean-curvature flow
is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time