Flat torus triangulations' space is homotopy equivalent to a torus.
problem Proving homotopy equivalence of triangulations of flat tori.
method Generalization of Tutte's embedding theorem for flat tori.
result Deformation space of geodesic triangulations is homotopy equivalent to a torus.
In this paper, we consider discrete groups in PGLd(R) acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for CAT(0) spaces.
New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
Proves uniqueness and existence of toric gravitational instantons.
problem Proves uniqueness and existence of four-dimensional asymptotically flat, Ricci-flat, toric gravitational instantons.
method Adapting black hole uniqueness theorems to a harmonic map formulation of Ricci-flat metrics with torus symmetry.
result Proves that instantons are uniquely characterised by their rod structure and that for every admissible rod structure, there exists a smooth instanton.
Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
Study spherical metrics on flat torus with cone singularities.
problem Existence and uniqueness of spherical metrics with specific cone angles.
method Analysis of Green function critical points and torus geometry.
result Existence of unique spherical metrics determined by torus geometry.
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient f…
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimens…
Given a closed flat 3-torus N, for each H>0 and each non-negative integer g, we obtain area estimates for closed surfaces with genus g and constant mean curvature H embedded in N. This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer g…
The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.
problem Geometric stability of the positive mass theorem and related conjectures.
method Analysis of harmonic maps to flat model spaces under integral curvature bounds.
result Upgrading harmonic maps to diffeomorphisms when conditions are met, proving quantitative closeness to model spaces.
The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.
problem Proving rigidity for minimal submanifolds in spheres with flat normal bundle.
method Explicit second-gap rigidity theorem for the squared norm of the second fundamental form.
result The theorem provides evidence for Chern's conjecture in higher codimension.
New findings on minimal isometric immersions of flat n-tori into spheres.
problem Conditions for minimal isometric immersions of flat n-tori into spheres.
method Analyzes rationality conditions and derives upper bounds for algebraic irrationality degree.
result Upper bound for algebraic irrationality degree of minimal isometric immersions is sharp and equals 4 for n=3.
The paper proves conditions for positive scalar curvature metrics on manifolds with incompressible hypersurfaces.
problem Conditions for the existence of metrics with positive scalar curvature on manifolds with incompressible hypersurfaces.
method Analyzing surgeries and applying the positive mass theorem with incompressible conditions.
result Establishes positive mass theorem with incompressible conditions for specific manifolds.
Upper bounds on revised first Betti number and torus stability for RCD spaces.
problem Bounding the revised first Betti number and stability of RCD spaces.
method Proving an upper bound on the rank of the abelianised revised fundamental group and establishing torus stability.
result Spaces with saturated upper bound on revised first Betti number are mGH-close to flat tori.
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.
Proves positive mass theorems for specific ALF and ALG manifolds.
problem Proving positive mass theorems for ALF and ALG manifolds.
method Reduces the problem to non-existence of positive scalar curvature metrics on closed manifolds.
result Shows that incompressible conditions for S1 and T2 are sufficient for nonnegativity of mass. New Finsler flow on 2-torus has chaotic dynamics.
problem Constructing chaotic dynamics on a 2-torus.
method Using Berger and Turaev's theorem, constructing a Finsler metric.
result Found a Finsler geodesic flow with positive metric entropy.
Origami creates flat torus models of any size.
problem Creating flat torus models of any size.
method Explicit origami folding instructions.
result Flat torus models of any size created.
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function al…
Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in C2. result Asymptotic counting result for tripods on the torus.
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…
New proof confirms flat equilateral torus is λ1-maximal.
problem Maximizing the first eigenvalue on flat tori.
method Combining El Soufi-Ilias-Ros's method and Bryant's result.
result Positive answer to Berger's isoperimetric problem.
Let X be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold X′, deformation equivalent to X, which is not an analytification of any projective variety, if and only if H0(X,Ω2)=0. Using this, we recover a recent theorem of Catanese and Demleitner, which stat…
We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature H in a space form with constant sectional curvature c. First we extend a theorem due to Defever when c=0 and show that there is no such hypersurface if H=0. Our main res…
Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.
problem Proving the geometric stability conjecture for scalar curvature on manifolds conformal to tori.
method Reduction via the Yamabe problem and handling sequences of manifolds conformal to flat tori or constant negative scalar curvature.
result Proves the geometric stability conjecture for certain sequences of manifolds conformal to flat tori.
Study shows convergence of certain metrics to flat torus.
problem Stability of metrics on three-torus with negative scalar curvature.
method Defined metrics and used Stern's inequality to show convergence.
result Subsequence of metrics converges to flat metric.
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
Universal triangulation for flat tori with 2434 triangles.
problem Embedding flat tori isometrically in 3D space.
method Adapted Burago and Zalgaller's proof for polyhedral surfaces, combined with Zalgaller's construction.
result A universal triangulation of 2434 triangles for any flat torus.
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
Model financial dynamics using 2-manifold geometries, revealing the torus as best for cyclical data.
problem Financial forecasting using complex market data.
method Embedding market data onto 2-manifolds (S2, R2, H2, T) guided by uniformization theorem, inferring latent curvature.
result The torus geometry best predicts cyclical financial data, aligning with IS-LM theory.
Proves product metrics are Yamabe metrics under small flat torus conditions.
problem Yamabe metrics on product spaces with small flat tori.
method Extends earlier results to Type~I and Type~II Yamabe constants, Q-curvature problems, and isoperimetric-ratio type problems. result Product metrics are Yamabe metrics for sufficiently small flat tori.
Proves Schoen's conjecture on tori with specific conditions.
problem Proving Schoen's conjecture on tori with non-negative scalar curvature.
method Uses weighted scalar curvature and the relative index theorem.
result If the fundamental group of the singular set is not surjective, the metric extends to a smooth flat metric.
The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…
Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
The article proves K5 and K3,3 are toroidal penny graphs.
problem Optimal sphere packing on torus.
method Analyzing connections between planar graphs, penny graphs, and toroidal penny graphs.
result K5 and K3,3 are toroidal penny graphs. Paper introduces flat-virtual knots and invariants for classical knots.
problem Constructing a map from classical knots to virtual knots.
method Definition of flat-virtual knots and invariants (Alexander-like polynomial, Kauffman bracket).
result Introduction of flat-virtual knots and their invariants.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.
problem Deforming foliated manifolds with flat leaves while maintaining curvature bounds.
method Collapsing a manifold with a closed flat regular Riemannian foliation, keeping curvature uniformly bounded.
result For compact, simply connected manifolds, foliations are given by torus actions.
By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subco…
We give an affirmative answer to the Halperin-Carlsson conjecture for the homologically injective torus actions on closed manifolds. This class contains holomorphic torus actions on compact Kahler manifolds, torus actions on compact Riemannian flat manifolds.
We address the question of determining the eigenvalues λ_n (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with n nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
We prove a formula for the determinant of Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray-Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also cons…
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…
The study classifies compact Cauchy horizons in vacuum spacetimes.
problem Classifying compact Cauchy horizons in vacuum spacetimes.
method Complete classification theorem based on topology and null generators.
result Different cases of compact Cauchy horizons with specific manifolds and spacetime properties.