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UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations Title The geometry of vacuum static spaces and deformations of scalar curvature Permalink https://escholarship.org/uc/item/55w1c98q Author Yuan, Wei Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ THE GEOMETRY OF VACUUM STATIC SPACES AND DEFORMATIONS OF SCALAR CURVATURE A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Wei Yuan June 2015 The Dissertation of Wei Yuan is approved: Professor Jie Qing, Chair Professor Richard Montgomery Professor Longzhi Lin Dean Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Wei Yuan 2015 Contents Abstract v Dedication vii Acknowledgments viii 1 Introduction and preliminaries 1 1.1 Introduction to vacuum static spaces . 1 1.2 Notations and conventions . 7 1.3 Statements of main theorems . 8 2 Classifications of vacuum static spaces 25 2.1 Bach flatness and vanishing of D-tensor . 25 2.2 The geometry of level surfaces . 31 2.3 3-dimensional cases . 40 3 Local rigidity phenomena 44 3.1 Local scalar curvature rigidity in general . 44 3.2 Local scalar curvature rigidity of space forms . 56 3.2.1 Euclidean spaces . 56 3.2.2 Hyperbolic spaces . 58 3.2.3 Hemispheres . 61 3.3 Conformal rigidity of vacuum static space . 62 4 Brown-York mass and compactly conformal deformations 68 4.1 Brown-York mass and conformal rigidity . 68 4.2 Non-rigidity phenomena . 74 Bibliography 87 iii A Variations of curvature 95 A.1 Metrics . 96 A.2 Christoffel symbols . 97 A.3 Riemannian curvature tensor . 98 A.4 Ricci tensor . 101 A.5 Scalar curvature . 104 B Conformal transformations of Riemannian metric 108 B.1 Metric . 109 B.2 Christoffel symbols . 109 B.3 Hessian and Laplacian . 110 B.4 Second fundamental form and mean curvature . 110 B.5 Riemann curvature tensor . 111 B.6 Ricci curvature tensor . 114 B.7 Scalar curvature . 114 B.8 Traceless Ricci tensor . 116 B.9 Schouten tensor . 117 B.10 Weyl tensor . 117 B.11 Cotten Tensor . 118 B.12 Bach tensor . 120 C Curvature in warped product metrics 123 C.1 Metric . 124 C.2 Christoffel symbols . 124 C.3 Hessian and Laplacian . 125 C.4 Second fundamental form and mean curvature . 127 C.5 Riemann curvature tensor . 128 C.6 Ricci curvature tensor . 130 C.7 Scalar curvature . 131 C.8 Traceless Ricci tensor . 132 C.9 Schouten tensor . 133 C.10 Weyl tensor . 134 C.11 Cotten Tensor . 136 iv Abstract The geometry of vacuum static spaces and deformations of scalar curvature by Wei Yuan In this dissertation we mainly study the geometric structure of vacuum static spaces and some related geometric problems. In particular, we have made progress in solving the classification problem raised in [29] of vacuum static spaces and in proving the conjecture made in [5] about manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3. We also extend the local scalar curvature rigidity result in [10] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [20]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [30]. As for generic Riemannian manifolds, we find a connection between Brown- York mass and the first Dirichlet Eigenvalue of a Schrödinger type operator. In v particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly conformal deformations which either increase or decrease scalar curva- ture. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question arises naturally in [20, 40]. vi To my parents, grannies and Barbara McDowell (1918 - 2012) vii Acknowledgments I would like to express my deepest appreciations to my adviser Professor Jie Qing in Department of Mathematics. It is my honor to be one of his Ph. D students. In the past seven years, I learned a lot from him. It includes how to be a excellent mathematician, a fantastic adviser and a great teacher, but most important of all, a nice person. Especially, I would like to thank him for giving me valuable suggestions and advise when I was in confusions of how to make choices in jobs and life. Without his support, I may not make it. Also I would like to thank my parents for their support and taking good care of everything in the family so that I can pursue my dream without worrying much. Although they never got any higher education, they know what does it mean to their son. Finally, I would like to thank Professor Richard Montgomery, Professor Justin Corvino, Professor Pengzi Miao, Professor Longzhi Lin, Dr. David DeConde, Dr. Yi Fang and Dr. Yueh-Ju Lin for their inspiring discussions and help in writing this dissertation. Especially for those long conversations with Dr. Yi Fang, they have provided me confidence and inspirations. viii Chapter 1 Introduction and preliminaries 1.1 Introduction to vacuum static spaces In 1915, A.Einstein published the article "Die Feldgleichungen der Gravita- tion" ([26]), in which he proposed a field equation for vacuum space-time: Gµν = κTµν; 1 where Gµν = Rµν − 2 Rgµν is the Einstein tensor, κ is a constant and Tµν is the Energy-Momentum-Stress tensor. It is the first time in the history people introduced a geometric theory for gravitation. Now this theory is popularly known as "general relativity". As one of the most successful theories of physics in the twenty century, gen- eral relativity has revealed the fundamental interplay between physics and the 1 geometry of space-time. During the past 100 years, it has been one of the most active research fields in both physics and mathematics. It has become one of the fundamental tools for people to get to know the past and the future of our uni- verse. Besides its essential role in the theoretical study, general relativity has also gained great success in engineering when applying to our daily life. For example, general relativistic effect has to be considered to reduce the systemic errors when engineers try to design the Global Positioning System (GPS)( for more details please refer to [45]). After being proposed, searching for different solutions to Einstein’s field equa- tions became one of the most important problems. The most obvious solution is the Minkowski’s space-time, which is the four- dimensional Euclidean space R4 equipped with Lorentzian metric 2 g = −dt + δR3 ; 3 where δR3 is the Euclidean metric on R . In fact, this is one of the solution to vacuum Einstein’s field equations: Rµν = 0; which corresponds to a flat space-time. The first non-trivial solution was published in 1916 by K.Schwartzchild ([53]), 2 which is now known as the Schwartzchild’s solution: −1 2GM 2 2GM 2 2 g = − 1 − dt + 1 − dr + r g 2 ; r r S 2 where gS2 is the canonical spherical metric on S . Other well-know solutions were founded later as time went by, such as de Sitter/anti-de Sitter space-times, Kerr space-time, Friedmann-Lemaître-Robertson- Walker space-time, etc. Among all those solutions, static space-times are special and important global solutions. The Minkowski space-time, Schwartzchild space-time and de Sitter/anti- de Sitter space-times are typical examples of static space-time. One of the com- mon features is that the time direction after a proper rescaling can be written as a Killing field, when evolving along the integral curve of which, the spatial slices remain invariant. This is the reason we call them "static" according to Newton’s viewpoint on space and time. When working in a more abstract frame, we can consider static space-times that carry a perfect fluid matter field as introduced in [32, 35]. One may include a cosmological constant to maintain mass-energy density to be nonnegative and consider a Lorentzian manifold M^ = R × M with a static space-time metric g^ = −f 2dt2 + g which satisfies the Einstein’s field equation 1 Ric^ − R^g^ + Λ^g = 8πGT; (1.1.1) 2 3 for the energy-momentum-stress tensor T = µf 2dt2 +pg of a perfect fluid, where f is independent of time and called a lapse function, µ and p are nonnegative, time- independent mass-energy density and pressure of the perfect fluid respectively and G is the gravitational constant. Here we take the speed of light to be c = 1. In fact, the energy density and pressure can be expressed in terms of the Riemannian metric g and cosmological constant Λ: R µ = g − Λ (1.1.2) 2 and n − 1 ∆ f n − 2 p = g − R + Λ; (1.1.3) n f 2(n − 1) g where ∆g and Rg are the Laplace-Beltrami operator and scalar curvature of g respectively.
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