The Schmidt metric is described for Lorentzian surfaces and its curvature is analyzed.
problem Analyzing the curvature of Lorentzian surfaces using the Schmidt metric.
method General form of the Schmidt metric for Lorentzian surfaces, Ricci scalar calculation.
result The Ricci scalar of the Schmidt metric in terms of the Lorentzian manifold's Ricci scalar.
We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.
Let Gg,b be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface of genus g with b boundary components. We describe the set Ag,b of all automorphisms of graphs in Gg,b showing that, up to suitable moves changing the graph within …
For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^{6g-6+3b} onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization…
The paper shows pure mapping class groups are linear in certain cases.
problem Understanding linearity of pure mapping class groups.
method Analyzing presentations of pure mapping groups and comparing them to braid groups and Artin groups.
result Pure mapping class groups are linear in specific cases.
Paper finds a normal generating set for Torelli group of non-orientable surfaces.
problem Finding a normal generating set for the Torelli group of non-orientable surfaces.
method Explicit construction using BSCC and BP maps.
result An explicit normal generating set for the Torelli group of genus-g compact non-orientable surfaces with b boundary components. The paper sets geometric lower bounds for low Steklov eigenvalues on manifolds.
problem Finding geometric lower bounds for low Steklov eigenvalues on manifolds.
method Using trace inequalities relating Steklov eigenvalues to Neumann eigenvalues of subdomains containing boundary collars.
result Geometric lower bounds for low Steklov eigenvalues, complementing earlier results.
Study essential surfaces in knot exteriors, proving some combinations are realizable.
problem Whether given genus, boundary components, and slope combinations are realizable in knot exteriors.
method Analyzing compact orientable essential surfaces in knot exteriors, proving existence for even boundary components.
result For even boundary components, there exist knots with essential surfaces of specified genus and slope.
The paper constructs finite generating sets for complex algebraic structures.
problem Finite generation of specific algebraic structures.
method Explicit construction of finite generating sets for γ2IAn and γ2Inb. result Explicit finite generating sets for γ2IAn and almost explicit for γ2Inb. Researchers prove abelianizations of specific groups are finitely generated.
problem Proving finitely generated nature of abelianizations of specific groups.
method New sufficient condition for modules over Laurent polynomial rings.
result Proves finitely generated nature of abelianizations (K3b)ab and [IO3,IO3]ab. Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.