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The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality
 J. DIFFERENTIAL GEOM
, 1998
"... In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announc ..."
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Cited by 201 (0 self)
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In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announced in [HI1]. Let M be a smooth Riemannian manifold of dimension n 2 with metric g = (g ij ). A classical solution of the inverse mean curvature flow is a smooth family x : N \Theta [0; T ] !M
Variational theory for the total scalar curvature functional for Riemannian metrics and related topics
 in Topics in Calculus of Variations (Montecatini
, 1987
"... The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We ..."
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Cited by 177 (2 self)
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The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss variational characterizations of constant curvalure m trics, and give a proof of 0bata's uniqueness theorem. Much of the material in this section can be found in papers of Berger Ebin [3], FischerMarsden [8], N. Koiso [14], and also in the recent book by A. Besse [4] where the reader will find additional references. In §2 we give a general discussion of the Yamabe problem and its resolution. We also give a detailed analysis of the solutions of the Yamabe equation for the product conformal structure on SI(T) x S~1(1), a circle of radius T crossed with a sphere of radius one. These exhibit interesting variational fea,tures uch a.s symmetry breaking and the existence of solutions with high Morse index. Since the time of the summer school in Montecatini, the beautiful survey paper of J. Lee and T. Parker [15] has appeared. This gives a detailed discussion of the
FOURMANIFOLDS WITHOUT EINSTEIN METRICS
 MATHEMATICAL RESEARCH LETTERS 3, 133–147 (1996)
, 1996
"... It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type, and the meth ..."
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Cited by 77 (14 self)
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It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type, and the method ofproofstems from SeibergWitten theory.
Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 61 (10 self)
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We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group H1. In H1, identified with the Euclidean space R 3, the pminimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the pmean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining function, and as a quantity in terms of calibration geometry. We also show that there are no closed, connected, C 2 smoothly embedded constant pmean curvature or pminimal surfaces of genus greater than one in the standard S 3. This fact
Local Estimates for a Class of Fully Nonlinear Equations Arising From Conformal Geometry
 Int. Math. Res. Not
, 2001
"... this paper, we are interested in a class of fully nonlinear differential equations related to the deformation of conformal metrics. Let (M; g 0 ) be a compact connected smooth Riemannian manifold of dimension n 3, and let [g 0 ] denote the conformal class of g 0 . The Schouten tensor of the metric ..."
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Cited by 52 (10 self)
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this paper, we are interested in a class of fully nonlinear differential equations related to the deformation of conformal metrics. Let (M; g 0 ) be a compact connected smooth Riemannian manifold of dimension n 3, and let [g 0 ] denote the conformal class of g 0 . The Schouten tensor of the metric g is defined as S g = 1 n \Gamma 2 ` Ric g \Gamma R g 2(n \Gamma 1) \Delta g ' ; where Ric g and R g are the Ricci tensor and scalar curvature of g respectively. This tensor is connected to the study of conformal invariants, in particular conformally invariant tensors and differential operators (e.g., see [6] and references therein). In [16], The following oe k scalar curvatures of g were considered by Viaclovsky in [16]: oe k (g) := oe k (g \Gamma1 \Delta S g ); where oe k is the kth elementary symmetric function, g \Gamma1 \DeltaS g is locally defined by (g \Gamma1 \DeltaS g ) i j = g ik (S g ) kj . When k = 1, oe 1 scalar curvature is just the scalar curvature R (upto a constant multiple). oe k can also be viewed as a function of the eigenvalues of symmetric matrices, that is a function in R n . According to G arding [7], \Gamma + k = f = ( 1 ; 2 ; \Delta \Delta \Delta ; n ) 2 R n j oe j () ? 0; 8j kg; is a natural class for oe k . A metric g is said to be in \Gamma + k if oe j (g)(x) ? 0 for j k and x 2 M . The case of k = 1, deforming scalar curvature R to a constant in its conformal class is known as the Yamabe problem, the final solution was obtained by Schoen in [12] (see also [1] and [15]). We refer [10] for the literature on Yamabe problem. There is a recent interest in deforming oe k scalar curvature in its conformal class. This type of problem was Date: August, 2001. 1991 Mathematics Subject Classification. [. Key words a...
On Conformally Kähler, Einstein Manifolds
, 2007
"... We prove that any compact complex surface with c1> 0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blowup CP2#2CP2 of the complex projective plane at two distinct points. 1 ..."
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Cited by 51 (11 self)
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We prove that any compact complex surface with c1> 0 admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blowup CP2#2CP2 of the complex projective plane at two distinct points. 1
Moduli spaces of critical Riemannian metrics in dimension four
"... Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can ..."
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Cited by 43 (2 self)
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Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. 1.
Weyl structures for parabolic geometries
 MATH. SCAND
, 2003
"... Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces G/P with G semisimple and P parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions ..."
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Cited by 42 (11 self)
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Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces G/P with G semisimple and P parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, closed and exact Weyl connections, and Rho–tensors, we characterize the classes of such objects, and we use the results to give a new description of the Cartan bundles and connections for all parabolic geometries.
Yamabe constants and the perturbed SeibergWitten equations
 Comm. Anal. Geom
, 1997
"... Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1 ..."
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Cited by 42 (9 self)
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Among all conformal classes of Riemannian metrics on CP2, that of the FubiniStudy metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the SeibergWitten equations, also yields new results on the total scalar curvature of almostKähler 4manifolds. 1
Kodaira dimension and the Yamabe problem
 Comm. Anal. Geom
, 1999
"... The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unitvolume constantscalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constantscalarcurvature metrics which are Yamabe minimizers, but this does not ..."
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Cited by 41 (4 self)
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The Yamabe invariant Y (M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unitvolume constantscalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers constantscalarcurvature metrics which are Yamabe minimizers, but this does not affect the sign of the answer.) If M is the underlying smooth 4manifold of a complex algebraic surface (M, J), it is shown that the sign of Y (M) is completely determined by the Kodaira dimension