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UNIQUENESS OF STATIC VACUUM EINSTEIN METRICS AND THE BARTNIK QUASILOCAL MASS
"... Abstract. We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed ‘geometric ’ or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasilocal mas ..."
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Abstract. We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed ‘geometric ’ or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasilocal mass. A variational characterization of Bartnik boundary data is also given. 1.
ON THE LINEAR STABILITY OF EXPANDING RICCI SOLITONS
"... Abstract. In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture ..."
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Abstract. In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are linearly stable. This paper makes progress towards that conjecture, showing that expanding Ricci solitons with bounded curvature (including solvsolitons) are linearly stable after extension by a Gaussian soliton. As in the previous work, the dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.
OPTIMAL TRANSPORTATION AND MONOTONIC QUANTITIES ON EVOLVING MANIFOLDS
, 908
"... Abstract. In this note we will adapt Topping’s Loptimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij, where Sij is a symmetric tensor field of (2,0)type on M. We extend some recent results of Topping, Lott and Bre ..."
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Abstract. In this note we will adapt Topping’s Loptimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij, where Sij is a symmetric tensor field of (2,0)type on M. We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List’s (and hence also of Perelman’s) Wentropy, and recover the monotonicity of Müller’s (and hence also of Perelman’s) reduced volume. 1.
(1.1) Ya(g,u,t) = − M
, 712
"... Abstract. In this note, we establish the first variation formula of the adjusted log entropy functionalYa introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity ofYa along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci f ..."
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Abstract. In this note, we establish the first variation formula of the adjusted log entropy functionalYa introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity ofYa along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci flow. Let (M n,g(t)) be a smooth family of Riemannian metrics on a closed manifold M n and suppose g(t) is a solution of Hamilton’s Ricci flow equation. In a recent interesting paper [14], R. Ye introduced a new entropy functional, the adjusted log entropy, as follows
ON THE LOWER SEMICONTINUITY OF THE ADM MASS
"... Abstract. The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger–Gromov co ..."
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Abstract. The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger–Gromov convergence (without any symmetry assumptions) for n = 3, and second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, for n ≥ 3. We also apply recent results of LeFloch and Sormani to deal with the rotationally symmetric case, with respect to a pointed type of intrinsic flat convergence. We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass. 1.
4 A NOTE ON LOWER DIAMETER BOUNDS FOR CLOSED DOMAIN MANIFOLDS OF SHRINKING RICCIHARMONIC SOLITONS
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