arXiv subjects

Dan A. Lee

Publications and source records attributed to Dan A. Lee.

23 records · Page 2Linked to original sources

arXiv2011

Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results, as discussed in [LS]

math.DG↗

arXiv2008

Covering spaces and Q-gradings on Heegaard Floer homology

Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This induces a relative Q-grading on \bigoplus_{t\in Spin^c(Y)} HF(Y,t). In this paper we describe an alternate construction of this relative Q-grading by studying the Heegaard Floer homology of covering spaces.

math.GT↗

arXiv2007

On the Riemannian Penrose inequality in dimensions less than 8

The Positive Mass Theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass. 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 surface, 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. In 1999, H. Bray extended this result to the general case of multiple black holes using a different technique. In this paper we extend Bray's technique to dimensions less than 8.

math.DG↗

arXiv2007

On the near-equality case of the Positive Mass Theorem

The Positive Mass Conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the Positive Mass Conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The Positive Mass Conjecture was proved by R. Schoen and S.-T. Yau for all manifolds of dimension less than 8, and it was proved by E. Witten for all spin manifolds. In this paper, we consider complete asymptotically flat manifolds of nonnegative scalar curvature that are also harmonically flat in an end. We show that, whenever the Positive Mass Theorem holds, any appropriately normalized sequence of such manifolds whose masses converge to zero must have metrics that are uniformly converging to Euclidean metrics outside a compact region. This result is an ingredient in a forthcoming proof, co-authored with H. Bray, of the Riemannian Penrose inequality in dimensions less than 8.

math.DG↗

arXiv2003

Connected sums of special Lagrangian submanifolds

Let $M_1$ and $M_2$ be special Lagrangian submanifolds of a compact Calabi-Yau manifold $X$ that intersect transversely at a single point. We can then think of $M_1\cup M_2$ as a singular special Lagrangian submanifold of $X$ with a single isolated singularity. We investigate when we can regularize $M_1\cup M_2$ in the following sense: There exists a family of Calabi-Yau structures $X_α$ on $X$ and a family of special Lagrangian submanifolds $M_α$ of $X_α$ such that $M_α$ converges to $M_1\cup M_2$ and $X_α$ converges to the original Calabi-Yau structure on $X$. We prove that a regularization exists in two key cases: (1) when the complex dimension of $X$ is three, $\Hol(X)=\SU(3)$, and $[M_1]$ is not a multiple of $[M_2]$ in $H_3(X)$, and (2) when $X$ is a torus with complex dimension at least three, $M_1$ is flat, and the intersection of $M_1$ and $M_2$ satisfies a certain angle criterion. One can easily construct examples of the second case, and thus as a corollary we construct new examples of non-flat special Lagrangian submanifolds of Calabi-Yau tori.

math.DG↗