Yau Conjecture Research Articles

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.

Branched covers of hyperbolic manifolds and harmonic maps

Let f : M → N be a homotopy equivalence between closed negatively curved manifolds. The fundamental existence results of Eells and Sampson [5] and uniqueness of Hartmann [15] and Al’ber [1] grant the existence of a unique harmonic map h homotopic to f . Based on the enormous success of the harmonic map technique, Lawson and Yau conjectured that the harmonic map h should be a diffeomorphism. This conjecture was proved to be false by Farrell and Jones [6] in every dimension in which exotic spheres exist. They constructed examples of homeomorphisms f : M → N between closed negatively curved manifolds for which f is not homotopic to a diffeomorphism. These counterexamples were later obtained also in dimension six by Ontaneda [21] and later generalized by Farrell, Jones and Ontaneda to all dimensions > 5 [8]. In fact, in [21] and [8] examples are given for which f is not even homotopic to a PL homeomorphism. The fact that f is not homotopic to a PL homeomorphism has several interesting strong consequences that imply certain limitations of well known powerful analytic ethods in geometry [9], [10], [11], [12] (see [13] for a survey). In all the examples mentioned above, one of the manifolds is always a closed hyperbolic manifold. Hence, both manifolds M and N have the homotopy type of a closed hyperbolic manifold (hence the homotopy type of a closed locally symmetric space). We call these examples of hyperbolic homotopy type. In [2], Ardanza also gave counterexamples to the Lawson–Yau conjecture. In his examples, the manifolds M and N are not homotopy equivalent to a closed locally symmetric space; in particular, they are not homotopy

On Jacobian Ideals Invariant by a Reducible 𝑠ℓ(2,𝐂) Action

This paper deals with a reducible s ℓ ( 2 , C ) s\ell (2, \mathbf {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 s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) acts on the formal power series ring via ( 0.1 ) (0.1) . Then I ( f ) = ( ℓ i 1 ) ⊕ ( ℓ i 2 ) ⊕ ⋯ ⊕ ( ℓ i s ) I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}}) modulo some one dimensional s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representations where ( ℓ i ) (\ell _{i}) is an irreducible s ℓ ( 2 , C ) s\ell (2, \mathbf {C}) representation of dimension ℓ i \ell _{i} or empty set and { ℓ i 1 , ℓ i 2 , … , ℓ i s } ⊆ { ℓ 1 , ℓ 2 , … , ℓ r } \{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\} . Unlike classical invariant theory which deals only with irreducible action and 1–dimensional representations, we treat the reducible action and higher dimensional representations succesively.

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact K\"ahler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.

Coordinate-free characterization of homogeneous polynomials with isolated singularities

The Durfee conjecture, proposed in 1978, relates two important invariants of isolated hypersurface singularities by a famous inequality; however, the inequality in this conjecture is not sharp. In 1995, Yau announced his conjecture which proposed a sharp inequality. The Yau conjecture characterizes the conditions under which an affine hypersurface with an isolated singularity at the origin is a cone over a nonsingular projective hypersurface; in other words, the conjecture gives a coordinate-free characterization of when a convergent power series is a homogeneous polynomial after a biholomorphic change of variables. In this paper, we have proved that if pg > 0, then 5!pg ≤ μ− p(v), p(v) = (v − 1) − v(v − 1) . . . (v − 4) and pg, μ and v are, respectively, the geometric genus, the Milnor number, and the multiplicity of the isolated singularity at the origin of a weighted homogeneous polynomial. As a consequence, we prove that the Yau conjecture holds for n = 5 if pg > 0. In the process, we have also defined yet another sharp upper bound for the number of positive integral points within a five-dimensional simplex.