Yau Conjecture Research Articles

Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (related to a cohomogeneity one point of view on a Yau conjecture)

Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (related to a cohomogeneity one point of view on a Yau conjecture)

On projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $\phi:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite \'etale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with the theory of holomorphic foliations. Moreover, we show that the universal cover of $X$ is biholomorphic and isometric to the product $\mathbb{C}^m \times F$ of the universal cover $\mathbb{C}^m$ of $Y$ with a flat metric and the rationally connected fiber $F$ with a K\"ahler metric whose holomorphic sectional curvature is semi-positive.

The embedded Calabi–Yau conjecture for finite genus

Suppose that M is a complete, embedded minimal surface in R3 with an infinite number of ends, finite genus, and compact boundary. We prove that the simple limit ends of M have properly embedded representatives with compact boundary, genus zero, and constrained geometry. We use this result to show that if M has at least two simple limit ends, then M has exactly two simple limit ends. Furthermore, we demonstrate that M is properly embedded in R3 if and only if M has at most two limit ends if and only if M has a countable number of limit ends.

Planck-scale number of nodal domains for toral eigenfunctions

We study the number of nodal domains in balls shrinking slightly above the Planck scale for “generic” toral eigenfunctions. We prove that, up to the natural scaling, the nodal domains count obeys the same asymptotic law as the global number of nodal domains. The proof, on one hand, uses new arithmetic information to refine Bourgain's de-randomisation technique at Planck scale. And on the other hand, it requires a Planck scale version of Yau's conjecture which we believe to be of independent interest.

The Lieb-Yau conjecture for ground states of pseudo-relativistic Boson stars

It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass N>0 satisfies N<N⁎, where the finite constant N⁎ is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each N<N⁎. In this paper, we prove that the above uniqueness conjecture holds for the particular case where N>0 is small enough.

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen-Cahn min-max constructions were recently carried out by Guaraco and Gaspar-Guaraco. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie-Marques-Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every $p$ = 1, 2, 3, ..., a two-sided embedded minimal surface with Morse index $p$ and area $p^{1/3}$, as conjectured by Marques-Neves.

The genus-one global mirror theorem for the quintic -fold

We prove the genus-one restriction of the all-genus Landau–Ginzburg/Calabi–Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This gives the first evidence supporting the higher-genus Landau–Ginzburg/Calabi–Yau correspondence for the quintic $3$ -fold, and exhibits the first instance of the ‘genus zero controls higher genus’ principle, in the sense of Givental’s quantization formalism, for non-semisimple cohomological field theories.

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress. In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.

Yau's first eigenvalue conjecture in the isoparametric case

In this paper, we first survey the development of S. T. Yau's first eigenvalue conjecture and its complete solution in the isoparametric case,and introduce its relationship with Lawson's conjecture. Furthermore, we calculate the first eigenvalues of the isoparametric focal submanifolds (minimal submanifolds of the sphere),and propose the generalization of Yau's conjecture for the higher codimension case.

Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture

Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$ there exists $c>0$ such that for any Laplace eigenfunction $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$, the following inequality holds: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda =0\})$.

Derivations of the moduli algebras of weighted homogeneous hypersurface singularities

Let R=C[x1,x2,⋯,zn]/(f) where f is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then R, and Der(R), the Lie algebra of derivations on R, are graded. It is well-known that Der(R) has no negatively graded component [10]. J. Wahl conjectured that the above fact is still true in higher codimensional case provided that R=C[x1,x2,⋯,xn]/(f1,f2,⋯,fm) is an isolated, normal and complete intersection singularity and f1,f2,⋯,fm are weighted homogeneous polynomials with the same weight type (w1,w2,⋯,wn). On the other hand the first author Yau conjectured that the moduli algebra A(V)=C[x1,x2,⋯,xn]/(∂f/∂x1,⋯,∂f/∂xn) has no negatively weighted derivations where f is a weighted homogeneous polynomial defining an isolated singularity at the origin. Assuming this conjecture has a positive answer, he gave a characterization of weighted homogeneous hypersurface singularities only using the Lie algebra Der(A(V)) of derivations on A(V). The conjecture of Yau can be thought as an Artinian analogue of J. Wahl's conjecture. For the low embedding dimension, the Yau conjecture has a positive answer. In this paper we prove this conjecture for any high-dimensional singularities under the condition that the lowest weight is bigger than or equal to half of the highest weight.

On large families of bundles over algebraic surfaces

The aim of this note is to construct sequences of vector bundles with unbounded rank and discriminant on an arbitrary algebraic surface. This problem, on principally polarized abelian varieties with cyclic Neron–Severi group generated by the polarization, was considered by Nakashima in connection with the Douglas–Reinbacher–Yau conjecture on the Strong Bogomolov Inequality. In particular we show that on any surface, the Strong Bogomolov Inequality S B I l is false for all l > 4 .

Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength like radius. Combined with Yau's conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semi-classical limit. We also obtain results that link the size of the nodal set to the growth of solutions of planar Schr\"odinger equations with small potential.

Isoparametric foliation and Yau conjecture on the first eigenvalue, II

This is a continuation of [23], which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g=4 distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with g=6, the present paper obtains estimates on all the eigenvalues, among others, giving an affirmative answer in one case to the problem posed in [23], which may be regarded as a generalization of Yau's conjecture. In two of the four unsettled cases in [23] for focal submanifolds M1 of OT-FKM-type, we prove the first eigenvalues to be their respective dimensions.

Isoparametric functions and related topics

The famous Yau conjecture asserts that the rst eigenvalue of every closed minimal embedded hypersurface in the unit sphere is just its dimension. Over several decades, research on the eigenvalues of the Laplace operator has always been a core issue in the study of geometry, many geometricians are committed to the study of Yau conjecture in recent years. However, the results that ever known are only some inequalities on the estimate of the rst eigenvalue. As a main result of this paper, we show that the rst eigenvalue of a closed minimal isoparametric hypersurface in the unit sphere is just its dimension. Furthermore, we show that under some dimensional conditions, the focal submanifolds of an isoparametric hypersurface in the unit sphere also have their dimensions as the rst eigenvalues. As the second main result of this paper, motivated by the famous Schoen-Yau-Gromov-Lawson surgery theory on scalar curvature, we make a surgery at the embedding hypersurface in a Riemannian manifold, constructing a new manifold with good geometry properties, which is called a double manifold. In particular, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold has complicated topological properties (but its characteristic classes can be precisely described) and carries a metric of positive scalar curvature, more importantly, the isoparametric foliation is kept. A more extensive de nition of Willmore surface is the Willmore submanifold, which is an extremal submanifold of Willmore functional in spheres. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. As the last two main results of this paper, by taking advantage of isoparametric functions of OT-FKM-type, we give a series of new examples of Willmore submanifolds in the unit sphere; hereafter, we give a uni ed geometric proof that both of focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. These new examples of Willmore submanifolds are all minimal in spheres, but in general not Einstein.

Isoparametric foliation and Yau conjecture on the first eigenvalue

A well-known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.

Sharp polynomial estimate of integral points in right-angled simplices

Characterization of homogeneous polynomials with isolated critical point at the origin follows from a study of complex geometry. Yau previously proposed a Numerical Characterization Conjecture. A step forward in solving this conjecture, the Granville–Lin–Yau Conjecture was formulated, with a sharp estimate that counts the number of positive integral points in n -dimensional ( n ⩾ 3 ) real right-angled simplices with vertices whose distances to the origin are at least n − 1 . The estimate was proven for n ⩽ 6 but has a counterexample for n = 7 . In this project we come up with an idea of forming a New Sharp Estimate Conjecture where we need the distances of the vertices to be n . We have proved this New Sharp Estimate Conjecture for n ⩽ 9 .

NON-PROPERLY EMBEDDED MINIMAL PLANES IN HYPERBOLIC 3-SPACE

In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with non-positive curvature. We show this result by constructing a non-properly embedded minimal plane in H3. Hence, this gives a counterexample to Calabi–Yau conjecture for embedded minimal surfaces in negative curvature case.

Classification of gradient space of dimension 8 by a reducible sℓ(2, C) action

This paper deals with a reducible sl(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sl(2,C) acts on the formal power series ring via (1.1). Then I(f) = (li1) ⊕ (li2) ⊕... ⊕ (lis) modulo some one dimensional sl(2,C) representations where (li) is an irreducible sl(2,C) representation of li dimension and {li1li2,...,lis} ⊆ {l1, l2...,lr}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.

Complete hypersurfaces in a Euclidean space [formula omitted] with constant mth mean curvature

In this paper, we consider n-dimensional oriented complete hypersurfaces with constant mth mean curvature of a Euclidean space R n + 1 . We characterize the hypersurface S k ( c ) × R n − k in a Euclidean space R n + 1 and show that generalized Yau conjecture is true for the class of oriented compact locally conformally flat hypersurfaces with positive constant mth mean curvature of a Euclidean space R n + 1 . When m = 2 , our results reduce to the results of Q.-M. Cheng [Q.M. Cheng, Complete hypersurfaces in a Euclidean space R n + 1 with constant scalar curvature, Indiana Univ. Math. J. 51 (2002) 53–68].