Annals of Global Analysis and Geometry最新文献

In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction of convex domains of a Riemannian manifold. Specifically, we focus on cases where the principal eigenfunction u is assumed to be log-concave and our primary goal is to obtain quantitative estimates for the Hessian of (log u).

In this work, we will establish new classification results concerning (lambda _1)-extremality for partial flag manifolds using a sufficient and necessary condition, in terms of Lie theoretic data, for a Kähler–Einstein metric over a generalized flag manifold to be a critical point for the functional that assigns for each Riemannian invariant Kähler metric its first positive eigenvalue of the associated Laplacian..

In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if a complete orientable minimal Plateau surface is stable and has quadratic area growth in (mathbb {R}^3 ), then it must be flat.

Using co-homogeneity one symmetries, we construct a two-parameter family of non-abelian (G_2)-instantons on every member of the asymptotically locally conical (mathbb {B}_7)-family of (G_2)-metrics on (S^3 times mathbb {R}^4 ), and classify the resulting solutions. These solutions can be described as perturbations of a one-parameter family of abelian instantons, arising from the Killing vector-field generating the asymptotic circle fibre. Generically, these perturbations decay exponentially to the model, but we find a one-parameter family of instantons with polynomial decay. Moreover, we relate the two-parameter family to a lift of an explicit two-parameter family of anti-self-dual instantons on Taub-NUT (mathbb {R}^4), fibred over (S^3) in an adiabatic limit.

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [19] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [7] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [42] under different assumptions.

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant via the (L^{frac{n}{2}})-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions 5 and 6.

Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. Lytchak and Petrunin established a similar result in dimension 3. For the higher dimensional result, the “generic set” is open and dense in the (C^{q})–topology for any (qge 2.) In Lytchak and Petrunin’s work, the “generic set” is a dense (G_{delta }) in the (C^{q})–topology for any (qge 2.) Here we show that the set of such metrics on a compact 3–manifold actually contains a set that is that is open and dense set in the (C^{q})–topology, provided (qge 3.)

We exploit a natural correspondence between holomorphic (2, 3, 5)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5 to present a new perspective in the study of symmetries of (2, 3, 5)-distributions. This leads to a number of new results in this classical subject, including an unexpected relation between the multiply-transitive families of models having 7- and 6-dimensional symmetries, and a one-to-one correspondence between equivalence classes of nontransitive (2, 3, 5)-distributions with 6-dimensional symmetries and nonhomogeneous nondegenerate Legendrian curves in ({{mathbb {P}}}^3). An ingredient for establishing the former is an explicit classification of homogeneous nondegenerate Legendrian curves in ({{mathbb {P}}}^3), which we present. Moreover, our approach gives a new perspective on exceptionality of the 3 : 1 ratio for two 2-spheres rolling on each other without twisting or slipping.

Given a noncompact disconnected periodic curve (Gamma ) of infinite length with two components and no self-intersection in (mathbb R^3), it is proved that there exists a noncompact simply connected periodic minimal surface spanning (Gamma ). As an application, it is shown that for any tetrahedron T with dihedral angles (le 90^circ ), there exist four embedded minimal annuli in T, which are perpendicular to (partial T) along their boundary. It is also proved that every Platonic solid of (mathbb R^3) contains a free boundary embedded minimal surface of genus zero.