arXiv research

Origami creates flat torus models of any size.

Flat torus triangulations' space is homotopy equivalent to a torus.

Counting tripods on a flat torus using lattice point counting.

New proof confirms flat equilateral torus is λ1-maximal.

Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.

Study shows convergence of certain metrics to flat torus.

Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.

Universal triangulation for flat tori with 2434 triangles.

Flow preserves volume on flat torus, converging to stable set.

Wave fronts on certain surfaces become dense.

Proves product metrics are Yamabe metrics under small flat torus conditions.

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.

The article proves $K5$ and $K_{3,3}$ are toroidal penny graphs.

Paper introduces flat-virtual knots and invariants for classical knots.

Study projective KLT varieties with projectively flat cotangent sheaves.

Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.

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…

New theorem on flat tori stability using harmonic maps and Ricci flow.

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.

Proves uniqueness and existence of toric gravitational instantons.

Study SKT and CYT manifolds with parallel Bismut torsion.

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti…

Exotic hypercomplex structures on a torus are proven to not exist.

Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.

Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ 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 ${\rm CAT}(0)$ spaces.

The paper develops a method to map knots in a cylinder to virtual-flat knots.

New findings on minimal isometric immersions of flat n-tori into spheres.

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homoge…

Upper bounds on revised first Betti number and torus stability for RCD spaces.

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat …

New metrics found on non-Kähler Calabi-Yau manifolds.

In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized com…

Formula found for probability of random triangles on flat tori being homotopically trivial.

Study spherical metrics on flat torus with cone singularities.

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…

Exposes two methods for constructing flat surfaces in 4D spaces.

It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial tor…

This study proves the local existence of a symplectic gradient flow on a flat torus.

Study shows surprising cobordism distances between certain torus knots.

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal str…

We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.

We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.

We prove that if the dimension of the first cohomology group of a $RCD^*(0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geomet…

New flat surfaces found in 3D sphere space.

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 $g$…