arXiv research

The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.

New findings on isospectral tori and harmonic maps between flat tori.

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq 3$. We will call these examples BSV-tori. In this note, we show that on a flat $6$-torus, all the orthogonal complex structures are either the complex tori or the BSV-to…

Study on proper-biharmonic flat tori in spheres with CMC conditions.

Flat tori found non-isometric pairs with identical Laplace eigenvalues.

Lower bounds for minimal hypersurfaces in flat tori.

We produce a new general family of flat tori in R^4, the first one since Bianchi's classical works in the 19th century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in S^3, we first present a global description of all isometric immersions of R^2 into R^4 with flat normal bundle.

Flat tori in 3-sphere have π extrinsic diameter under specific conditions.

The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.

Study shows tori metrics converging to flat under specific conditions.

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

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

Warped tori with almost non-negative scalar curvature converge to a flat torus.

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

The paper proves convergence of graphs of functions to flat tori under certain curvature conditions.

Study on complex tori foliations and flat geometries.

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…

The paper explores metrics and compactifications of flat tori.

Killing tensors on tori are shown to be polynomial in the metric and Killing vector fields.

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

Research shows how certain flat structures behave in specific convex domains.

Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note,…

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…

In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be 1 (mod 3). Next, we construct an explicit example in the higher codimensional ca…

In this paper, we consider new components of the key space of the Moduli space of minimal surfaces in flat 4-tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4-tori and obtain an element of the Moduli. In the process of the construction, we give an example of minimal surface…

Universal triangulation for flat tori with 2434 triangles.

In this paper we show that flat (m-1)-dimensional tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(m,Z) \SL(m,R)/SO(m). We also show that the dimension of the subspace of H_{m-1}(Γ\SL(m,R)/SO(m);Q) spanned by flat (m-1)-tori grows as one goes up in congruence covers.

A riemannian manifold is secure if the geodesics between any pair of points in the manifold can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. The conjecture claims, in particular, that a riem…

Flat minimal tori counterexamples refute Lu's second-gap conjecture.

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

Study on K3 surfaces with collapsing metrics and nilpotent structures.

Characterizes conformal classes of tori using differential geometry.

In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.

Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.

Origami creates flat torus models of any size.

Study on totally real flat minimal surfaces in quaternionic projective space.

Study solves equations on tori for Calabi-Yau problems.

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeo…

Geometrically interprets exact triangles of projectively flat bundles on complex tori.

Study of spectral geometry on noncommutative tori using functional metrics.

The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…

The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.

The paper proves a Connes trace theorem for curved noncommutative tori.

Study abelian varieties' Weil-Petersson metric asymptotics.

Compactifies moduli of abelian varieties and curves by attaching flat tori.

Characterizes Lamé equations with finite monodromy on flat tori.

New example of manifolds with monotonic heat kernels found.