Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates...

Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. Chern's conjecture for hypersurfaces in spheres...

Shiing-Shen Chern in 1955.). For manifolds of dimension 6 or higher the conjecture is open. An example of Robert Geroch had shown that the Chern–Gauss–Bonnet...

the same topic. Chern–Weil homomorphism Chern class Chern–Simons form Chern–Simons theory Chern's conjecture (affine geometry) Pontryagin number Pontryagin...

; Tang, Z. (2012). "Chern Conjecture and Isoparametric Hypersurfaces". Differential Geometry: Under the influence of S.S. Chern. Beijing: Higher Education...

Particularly well-known are a conjecture on existence of minimal hypersurfaces and on the spectral geometry of minimal hypersurfaces. In 1978, by studying the...

Poincaré and geometrization conjectures in 2003. Perelman was awarded a Millennium Prize for resolving the Poincaré conjecture but declined it, regarding...

a priori estimates for certain partial differential equations. In the 1970s, Shing-Tung Yau began working on the Calabi conjecture, initially attempting...

for: On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101–172. Compactness theorems for Kähler-Einstein...

of these balls are (n − 1)-dimensional spheres of radius ε {\displaystyle \varepsilon } ; their hypersurface measures ("areas") satisfy the following...

interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel...

Tung. Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), no. 3, 195–204. Rosenberg, Harold. Hypersurfaces of constant curvature in space...

between all (then) known Calabi–Yau compactifications in string theory; this partially supports a conjecture by Reid (1987) whereby conifolds connect all possible...

state of affairs was highly marked in the various cases of the Poincaré conjecture, in which four different proof methods are applied. The dimension of a manifold...

frame Hypersurface Induced metric Nash embedding theorem minimal surface Helicoid Catenoid Costa's minimal surface Hsiang–Lawson's conjecture Theorema...

curvature cannot be smoothly isometrically immersed as a hypersurface, and a theorem of Shiing-Shen Chern and Kuiper even says that any closed m-dimensional...

tensor. In part, this simplicity allows for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through...

especially in the compact setting, for global analytic results to be proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the...

and 2 (conic sections) occurs in Chapter 17, as does Mordell's conjecture. Siegel's theorem on integral points occurs in Chapter 28. Mordell's theorem...

Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach. In classical algebraic geometry, the...

of Hamilton's equations on tori Sergiu Klainerman, Null hypersurfaces and curvature estimates in general relativity Bruce Kleiner, Singular structure of...

This, in combination with the Goldberg–Kobayashi result, forms the final part of Yum-Tong Siu and Shing-Tung Yau's proof of the Frankel conjecture. Kobayashi...

generalization of a sphere). T-duality can be extended from circles to the three-dimensional tori appearing in this decomposition, and the SYZ conjecture states that...

characteristic classes for vector bundles that take values in cohomology, including Chern classes, Stiefel–Whitney classes, and Pontryagin classes. For each abelian...