scholarly journals Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

2017 ◽  
Vol 28 (2) ◽  
pp. 1245-1257
Author(s):  
Abraão Mendes
Keyword(s):  
Free Boundary ◽  
Convex Boundary ◽  
Strictly Convex ◽  
Boundary Surfaces

scholarly journals Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

2018 ◽  
Vol 168 (1) ◽  
pp. 29-41 ◽  
Author(s):  
JOONAS ILMAVIRTA ◽  
JERE LEHTONEN ◽  
MIKKO SALO
Keyword(s):  
Higher Dimensions ◽  
Uniqueness Result ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Two Dimensional ◽  
Local Uniqueness ◽  
Piecewise Constant ◽  
Convex Boundary ◽  
X Ray ◽  
Strictly Convex

AbstractWe show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

scholarly journals Marked boundary rigidity for surfaces

2016 ◽  
Vol 38 (4) ◽  
pp. 1459-1478 ◽  
Author(s):  
COLIN GUILLARMOU ◽  
MARCO MAZZUCCHELLI
Keyword(s):  
Riemannian Metrics ◽  
Compact Surface ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Convex Boundary ◽  
Strictly Convex ◽  
Negatively Curved ◽  
Boundary Rigidity ◽  
Boundary Distance

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group

2020 ◽  
Vol 31 (11) ◽  
pp. 2050086
Author(s):  
Ezequiel Barbosa ◽  
Farley Santana ◽  
Abhitosh Upadhyay
Keyword(s):  
Free Boundary ◽  
Convex Domain ◽  
Lie Group ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Connected Components ◽  
Geodesic Ball ◽  
Invariant Metric ◽  
Strictly Convex

Let [Formula: see text] be a three-dimensional Lie group with a bi-invariant metric. Consider [Formula: see text] a strictly convex domain in [Formula: see text]. We prove that if [Formula: see text] is a stable CMC free-boundary surface in [Formula: see text] then [Formula: see text] has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 73–479] for the case where [Formula: see text] and by R. Souam for the case where [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], excluding the possibility of [Formula: see text] having three boundary components. Besides [Formula: see text] and [Formula: see text], our result also apply to the spaces [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. When [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], we obtain that if [Formula: see text] is stable then [Formula: see text] is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

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