In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers proper... more In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.
Oscillations of Neutral Functional Differential Equations
Acta Mathematica Scientia, 1992
In this paper, we obtain some sufficient conditions for the oscillation and nonoscillation of som... more In this paper, we obtain some sufficient conditions for the oscillation and nonoscillation of some first order linear neutral functional differential equations. A mistake in the literature is also presented at the end of the paper.
ON SOLITARY WARE SOLUTIONS TO A GENERALIZED KdV EQNATION
Chinese Physics, 1993
The deformation theory, in Ref [5],of travelling wave solutions to a generalized KdV equation is ... more The deformation theory, in Ref [5],of travelling wave solutions to a generalized KdV equation is modified. The necessary and sufficient condition for a relevant algebraic equation to possess nonzero real root is presented and thus an error of analyses in Ref. [5] is pointe-dont. Finally an explicit formula of the solitary wave solutions generated by deformetion theory is obtained directly from the generalized KdV equation itselp.
Scalar Curvatures on Noncompact Riemann Manifolds
Chinese Annals of Mathematics, 1995
The author obtains some theorems for a function to be t he scalar curwture of some complete confo... more The author obtains some theorems for a function to be t he scalar curwture of some complete conformal metric of a noncompact complete Riemann manifold, and also presents a kind of manifolds on which Yamabe problem is unsolvable.
On the Morse index with constraints: An abstract formulation
Journal of Mathematical Analysis and Applications, Oct 1, 2023
Conformal Deformation of Complete Surface with Negative Curvature
Chinese Annals of Mathematics, 1997
The author considers the problem of deforming the metric on a complete negatively curved surface ... more The author considers the problem of deforming the metric on a complete negatively curved surface conformal to another metric whose Gauss curvature is nonpositive.
Chinese Annals of Mathematics, Series B, Jul 1, 2003
Theorems of Barth-Lefschetz type describe restrictions on the topology of varieties of small codi... more Theorems of Barth-Lefschetz type describe restrictions on the topology of varieties of small codimension. R. Schoen and J. Wolfson, using Morse theory on a path space, have described a technique to prove theorems of this kind for complex submanifolds of Kähler manifolds of non-negative holomorphic bisectional curvature. In this paper this program is carried out for the compact Hermitian symmetric spaces. The key technical point is to define and compute an invariant, called the complex positivity, that measures the ''amount'' of positive curvature, in a suitable sense.
In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a va... more In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC hypersurfaces, capillary surfaces in a ball, and critical points of type-II partitioning. In this paper, we study the index and nullity of a symmetric bounded bilinear form in a Hilbert space. The main results determine precisely how these notions change when restricting to a subspace of a finite codimension.
This paper concerns some stability properties of higher dimensional catenoids in R n+1 with n ≥ 3... more This paper concerns some stability properties of higher dimensional catenoids in R n+1 with n ≥ 3. We prove that higher dimensional catenoids have index one. We use δ-stablity for minimal hypersurfaces and show that the catenoid is 2 n-stable and a complete 2 n-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegativ... more We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero Gauss-Kronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).
Let M 4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n... more Let M 4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n+2). We get a sharp estimate for the first eigenvalue λ 1 (M) of the Laplacian which is λ 1 (M) ≤ (2n+1) 2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QH n and N is a compact quotient of the generalized Heisenberg group. When λ 1 (M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.
In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegativ... more We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero Gauss-Kronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).
Proceedings of the American Mathematical Society, Oct 1, 2009
This paper concerns some stability properties of higher dimensional catenoids in ℝ +1 with ≥ 3. W... more This paper concerns some stability properties of higher dimensional catenoids in ℝ +1 with ≥ 3. We prove that higher dimensional catenoids have index one. We use-stablity for minimal hypersurfaces and show that the catenoid is 2-stable and that a complete 2-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.
In this article, we provide some volume growth estimates for complete noncompact gradient Ricci s... more In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. We prove a sharp volume growth estimate for complete noncompact gradient shrinking Ricci soliton. Moreover, we provide upper bound volume growth estimates for complete noncompact quasi-Einstein manifolds with λ = 0. In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with λ < 0 and µ ≤ 0 have at most exponential volume growth.
Let (M, g, f) be a 4-dimensional complete noncompact gradient shrinking Ricci soliton with the eq... more Let (M, g, f) be a 4-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇^2f=λ g, where λ is a positive real number. We prove that if M has constant scalar curvature S=2λ, it must be a quotient of 𝕊^2×ℝ^2. Together with the known results, this implies that a 4-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton R^4, S^2×R^2 or S^3×R.
Let M 4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n... more Let M 4n be a complete quaternionic Kähler manifold with scalar curvature bounded below by −16n(n+2). We get a sharp estimate for the first eigenvalue λ 1 (M) of the Laplacian which is λ 1 (M) ≤ (2n+1) 2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QH n and N is a compact quotient of the generalized Heisenberg group. When λ 1 (M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.
In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Hilário Alencar, Gregório Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.
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