A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

1982/06/01 by Peter Li, Shing-Tung Yau, Shing–Tung Yau · 53 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Advanced Differential Equations and Dynamical Systems

paper · doi:10.1007/bf01399507

Cited by

• Extremal Metric for the First Eigenvalue on a Klein Bottle
• Volume gap for minimal submanifolds in spheres
• Generalizations of the Kolmogorov–Barzdin embedding estimates
• Reflective and quasi-reflective Bianchi groups
• Willmore submanifolds in a sphere
• Compact stable constant mean curvature surfaces in the Berger spheres
• Extremal metric for the first eigenvalue on a Klein bottle
• Finiteness of arithmetic Kleinian reflection groups
• Spectral estimates on 2-tori
• A disc maximizes Laplace eigenvalues among isoperimetric surfaces of revolution
• Free boundary and capillary minimal surfaces in spherical caps II: Low energy
• Conformal upper bounds for the first eigenvalue of the p-Laplacian
• A linear lower bound on the gonality of modular curves
• Minimal immersions of compact bordered Riemann surfaces with free boundary
• Willmore inequality on hypersurfaces in hyperbolic space
• Berger inequality for Riemannian manifolds with an upper sectional curvature bound
• Classification of Willmore 2-spheres in Sⁿ
• Minmax Hierarchies, Minimal Fibrations and a PDE based Proof of the\n Willmore Conjecture
• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• Explicit lower bounds on the modular degree of an elliptic curve
• Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton)
• Uniformization of surfaces with boundary and the application to the triple junction surfaces with negative Euler characteristic
• The isoperimetric problem for Lens spaces
• Finiteness of arithmetic hyperbolic reflection groups
• Existence of constant mean curvature 2-spheres in Riemannian 3-spheres
• Estimates for eigenvalues of the Paneitz operator
• Willmore orbits for isometric Lie actions
• Greatest least eigenvalue of the Laplacian on the Klein bottle
• Global minimizers for the doubly-constrained Helfrich energy: the\n axisymmetric case
• Existence of minimizing Willmore surfaces of prescribed conformal class
• The weighted isoperimetric inequality and Sobolev inequality outside convex sets
• Extrinsic systole of Seifert surfaces and distortion of knots
• Moebius geometry of submanifolds in 𝕊 n
• Graphical Willmore Problems with Low-Regularity Boundary and Dirichlet Data
• Second Eigenvalue Estimates and Spectral Rigidity for Submanifolds of Compact Rank-One Symmetric Spaces
• Two-dimensional curvature functionals with superquadratic growth
• Spectral properties of bipolar minimal surfaces in S⁴
• Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow
• Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d
• Eigenvalue upper bounds for the magnetic Schroedinger operator
• The index of the cosmological horizon and the area-charge-inequality
• Second Variation Formula for Eigenvalue Functionals on Surfaces
• The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture
• Maximization of the first nontrivial eigenvalue on the surface of genus two
• A new family of minimal surfaces of even genus in the three-dimensional sphere
• The first Steklov eigenvalue, conformal geometry, and minimal surfaces
• Conformal upper bounds for the volume spectrum
• On stable constant mean curvature surfaces in S2 X R and H2 X R
• A note on singularities in finite time for the constrained Willmore flow
• Variational problems in the theory of hydroelastic waves. [europepmc]
• Li-Yau inequalities for the Helfrich functional and applications. [europepmc]
• Optimal metrics for the first curl eigenvalue on 3-manifolds. [europepmc]
• A Sharp Quantitative Alexandrov Inequality and Applications to Volume Preserving Geometric Flows in 3D. [europepmc]