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Proof of the Riemannian Penrose inequality using the positive mass theorem
 MR MR1908823 (2004j:53046) MATHEMATICAL GENERAL RELATIVITY 73
"... We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature which contain minimal sphe ..."
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Cited by 119 (14 self)
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We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3manifold as a totally geodesic submanifold of a spacetime in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a mass A/16π, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass (also called the ADM mass). The Riemannian Penrose Conjecture then states that the total mass of an asymptotically flat 3manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3metric to a Schwarzschild 3metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the Positive Mass Theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black hole in a Schwarzschild metric, the Riemannian Penrose Conjecture follows. We also refer the reader to the beautiful work of Huisken and Ilmanen [30], who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole. In Sections 1 and 2, we motivate the problem, discuss important quantities like total mass and horizons of black holes, and state 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.
Prescribing scalar curvature on Sn and related problems
 319–410. GEOMETRY 19
, 1995
"... on the occasion of his 70th birthday This is a sequel to [30], which studies the prescribing scalar curvature problem on S". First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and ..."
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Cited by 56 (10 self)
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on the occasion of his 70th birthday This is a sequel to [30], which studies the prescribing scalar curvature problem on S". First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lowerdimensional cases n = 2,3. Next we study the problem for n 2 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is p E (n 2,n). The key point there is that for the class of K mentioned above we have completed Lm apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is = n 2, the L " estimates for solutions fail in general. In fact, two or more blowup points occur.
Positive mass theorem on manifolds admitting corners along a hypersurface
 EFFECT OF BOUNDARY GEOMETRY ON ADM MASS 13
"... We study a class of nonsmooth Riemannian manifolds with a weak regularity assumption across a hypersurface Σ. We first give an approximation scheme to mollify the metric. Then we prove the Positive Mass Theorem on these manifolds under a geometric boundary condition for the metrics separated by Σ. ..."
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We study a class of nonsmooth Riemannian manifolds with a weak regularity assumption across a hypersurface Σ. We first give an approximation scheme to mollify the metric. Then we prove the Positive Mass Theorem on these manifolds under a geometric boundary condition for the metrics separated by Σ. 1 Introduction and Statement of Results The wellknown Positive Mass Theorem in general relativity was first proved by R. Schoen and S.T. Yau in [8] for smooth asymptotically flat manifolds with nonnegative scalar curvature. It is interesting to know on what kind of nonsmooth Riemannian manifolds their techniques and results can be generalized.
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 the Riemannian Penrose inequality in dimensions less than eight
 DUKE MATHEMATICAL JOURNAL
, 2009
"... The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. Mo ..."
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Cited by 46 (7 self)
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The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole (see [HI]). In 1999, Bray extended this result to the general case of multiple black holes using a different technique (see [Br]). In this article, we extend the technique of [Br] to dimensions less than eight. Part of the argument is contained in a companion article by Lee [L]. The equality case of the theorem requires the added assumption that the manifold be spin.
Blowup phenomena for the Yamabe equation
 J. Amer. Math. Soc
, 2008
"... Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to ..."
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Cited by 43 (8 self)
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Abstract. Let (M, g) be compact Riemannian manifold of dimension n ≥ 3. A wellknown conjecture states that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M, g) is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions n ≥ 52. 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
The Yamabe problem for higher order curvatures
, 2005
"... Let M be a compact Riemannian manifold of dimension n. The kcurvature, for k = 1, 2, · · · , n, is defined as the kth elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The kYamabe problem is to prove the existence of a conformal metric whose kcurvature is a constant ..."
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Cited by 39 (7 self)
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Let M be a compact Riemannian manifold of dimension n. The kcurvature, for k = 1, 2, · · · , n, is defined as the kth elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The kYamabe problem is to prove the existence of a conformal metric whose kcurvature is a constant. When k = 1, it reduces to the wellknown Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the kYamabe problem was recently proved by Gursky and Viaclovsky for k> n. In this 2 paper we prove the existence of solutions for the remaining cases 2 ≤ k ≤ n, assuming that 2 the equation is variational.
Ricci Curvature, Minimal Volumes, and SeibergWitten Theory
, 2000
"... We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces ..."
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Cited by 37 (2 self)
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We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4manifold minimizes volume among all metrics satisfying a pointwise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new nonexistence results for Einstein metrics.