5,019 research outputs found
Rigidity of generalized Thurston's sphere packings on 3-dimensional manifolds with boundary
Motivated by Guo-Luo's generalized circle packings on surfaces with boundary
\cite{GL2}, we introduce the generalized Thurston's sphere packings on
3-dimensional manifolds with boundary. Then we investigate the rigidity of the
generalized Thurston's sphere packings. We prove that the generalized
Thurston's sphere packings are locally determined by the combinatorial scalar
curvatures. We further prove the infinitesimal rigidity that the generalized
Thurston's sphere packings can not be deformed while keeping the combinatorial
Ricci curvatures fixed.Comment: arXiv admin note: text overlap with arXiv:2309.0120
Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces
Inversive distance circle packings introduced by Bowers-Stephenson are
natural generalizations of Thurston's circle packings on surfaces. To find
piecewise Euclidean metrics on surfaces with prescribed combinatorial
curvatures, we introduce the combinatorial Calabi flow, the fractional
combinatorial Calabi flow and the combinatorial -th Calabi flow for the
Euclidean inversive distance circle packings. Due to the singularities possibly
developed by these combinatorial curvature flows, the longtime existence and
convergence of these combinatorial curvature flows have been a difficult
problem for a long time. To handle the potential singularities along these
combinatorial curvature flows, we do surgery along these flows by edge flipping
under the weighted Delaunay condition. Using the discrete conformal theory
recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics
on surfaces, we prove the longtime existence and global convergence for the
solutions of these combinatorial curvature flows with surgery. This provides
effective algorithms for finding piecewise Euclidean metrics on surfaces with
prescribed combinatorial curvatures
A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II
In this paper, we study a natural discretization of the smooth Gaussian
curvature on surfaces, which is defined as the quotient of the angle defect and
the area of a geodesic disk at a vertex of a polyhedral surface. It is proved
that each decorated piecewise Euclidean metric on surfaces with nonpositive
Euler number is discrete conformal to a decorated piecewise Euclidean metric
with this discrete curvature constant. We further investigate the prescribing
combinatorial curvature problem for a parametrization of this discrete
curvature and prove some Kazdan-Warner type results. The main tools are
Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean
metrics on surfaces and variational principles with constraints
On the classification of discrete conformal structures on surfaces
Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced
the discrete conformal structures on surfaces in an axiomatic approach and
studied its classification. In this paper, we give a full classification of the
discrete conformal structures on surfaces, which completes Glickenstein-Thomas'
classification. As a result, we find some new classes of discrete conformal
structures on surfaces, including some of the generalized circle packing
metrics introduced by Guo-Luo \cite{GL2}. The relationships between the
discrete conformal structures on surfaces and the 3-dimensional hyperbolic
geometry are also discussed
Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary
Motivated by Guo-Luo's generalized circle packings on surfaces with boundary
\cite{GL2}, we introduce the generalized sphere packings on 3-dimensional
manifolds with boundary. Then we investigate the rigidity of the generalized
sphere packing metrics. We prove that the generalized sphere packing metric is
determined by the combinatorial scalar curvature. To find the hyper-ideal
polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial
scalar curvature, we introduce the combinatorial Ricci flow and combinatorial
Calabi flow for the generalized sphere packings on 3-dimensional manifolds with
boundary. Then we study the longtime existence and convergence for the
solutions of these combinatorial curvature flows.Comment: To appear in CVPD
A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces
In this paper, we introduce a new discretization of the Gaussian curvature on
surfaces, which is defined as the quotient of the angle defect and the area of
some dual cell of a weighted triangulation at the conic singularity. A discrete
uniformization theorem for this discrete Gaussian curvature is established on
surfaces with non-positive Euler number. The main tools are Bobenko-Lutz's
discrete conformal theory for decorated piecewise Euclidean metrics on surfaces
and variational principles with constraints
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