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L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Ricci flow and nonnegativity of sectional curvature
 11 (2004), 883–904, MR2106247, Zbl pre02152382
"... Abstract. In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional ..."
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Cited by 8 (5 self)
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Abstract. In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow.
Uniformization of conformally finite Riemann surfaces by the Ricci flow
, 2007
"... In this paper we give a new proof of the uniformization of conformally finite Riemann surface of negative Euler characteristic by the Ricci flow. Specifically, we will consider the normalized Ricci flow ∂gij = −(R −r)gij on a com∂t plete surface (M, g0) where M = N ∪L1 · · ·∪Lk, N is a compact man ..."
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Cited by 3 (1 self)
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In this paper we give a new proof of the uniformization of conformally finite Riemann surface of negative Euler characteristic by the Ricci flow. Specifically, we will consider the normalized Ricci flow ∂gij = −(R −r)gij on a com∂t plete surface (M, g0) where M = N ∪L1 · · ·∪Lk, N is a compact manifold and L1,..., Lk are the hyperbolic ends. Moreover, each (Li, g0) is asymptotically close to the hyperbolic cusp, a metric of constant curvature −1. We prove that the flow g(t) exponentially converges to the metric of constant negative curvature (−1). 1
Vol. 97, No. 2 DUKE MATHEMATICAL JOURNAL © 1999 VOLUME GROWTH, GREEN’S FUNCTIONS, AND PARABOLICITY OF ENDS
"... (n ≥ 2) without boundary. We fix a point o ∈ M once and for all, and we write x  =d(x,o) for the distance between points x and o and write V(r) =B(o,r) for the volume of the geodesic ball of radius r centered at o. The purpose of this paper is to find conditions on M in order to characterize the ..."
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(n ≥ 2) without boundary. We fix a point o ∈ M once and for all, and we write x  =d(x,o) for the distance between points x and o and write V(r) =B(o,r) for the volume of the geodesic ball of radius r centered at o. The purpose of this paper is to find conditions on M in order to characterize the existence of (positive)
ON MINIMAL HYPERSURFACES WITH FINITE HARMONIC INDICES
"... We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in R n+1 (n ≥ 3) and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Eac ..."
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We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in R n+1 (n ≥ 3) and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends. 1.
RICCI FLOW AND NONNEGATIVITY OF CURVATURE
, 2003
"... Abstract. In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimensi ..."
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Abstract. In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectional curvature, even though the nonnegativity of the sectional curvature was proved to be preserved by Hamilton in dimension three. The example is the first of this type. This fact is proved through a general splitting theorem on the complete family of metrics with nonnegative sectional curvature, deformed by the Ricci flow. §0 Introduction. The Ricci flow has been proved to be an effective tool in the study of the geometry and topology of manifolds. One of the good properties of the Ricci flow is that it preserves the ‘nonnegativity ’ of the curvature. In dimension three, Hamilton [ H1] proves that on compact manifolds the Ricc flow preserves the nonnegativity of the Ricci curvature and
The structure of stable minimal hypersurfaces in IR n+1
, 1997
"... We provide a new topological obstruction for complete stable minimal hypersurfaces in IR n+1. For n ≥ 3, we prove that a complete orientable stable minimal hypersurface in IR n+1 cannot have more than one end by showing the existence of a bounded harmonic function based on the Sobolev inequality for ..."
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We provide a new topological obstruction for complete stable minimal hypersurfaces in IR n+1. For n ≥ 3, we prove that a complete orientable stable minimal hypersurface in IR n+1 cannot have more than one end by showing the existence of a bounded harmonic function based on the Sobolev inequality for minimal submanifolds [MS] and by applying the Liouville theorem for harmonic functions due to SchoenYau [SY]. 1