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Abstract We develop new tools to analyze the complexity of the conjugacy equivalence relation , whenever is a left‐orderable group. Our methods are used to demonstrate nonsmoothness of for certain groups of dynamical origin, such as certain amalgams constructed from Thompson's group . We also initiate a systematic analysis of , where is a 3‐manifold. We prove that if is not prime, then is a universal countable Borel equivalence relation, and show that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . We also prove that if is the complement of a nontrivial knot in then is not smooth, and show how determining smoothness of for all knot manifolds is related to the L‐space conjecture.
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Regularity of viscosity solutions of the σk$\sigma _k$‐Loewner–Nirenberg problem
Abstract We study the regularity of the viscosity solution of the ‐Loewner–Nirenberg problem on a bounded smooth domain for . It was known that is locally Lipschitz in . We prove that, with being the distance function to and sufficiently small, is smooth in and the first derivatives of are Hölder continuous in . Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of and its covariant derivatives and vanishes if and only if is smooth in . Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when contains more than one connected components, is not differentiable in .
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- Award ID(s):
- 2000261
- PAR ID:
- 10420786
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 127
- Issue:
- 1
- ISSN:
- 0024-6115
- Format(s):
- Medium: X Size: p. 1-34
- Size(s):
- p. 1-34
- Sponsoring Org:
- National Science Foundation
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