arXiv research

We show numerically that any of the constant mean curvature tori first found by Wente must have index at least eight.

We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply.

Isothermic tori with one planar curvature line found and characterized.

Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.

Constructs surfaces with conical singularities using variational methods.

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Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.

We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.

Using the functional associated with the optimal Wente inequality for pairs of functions on 2-dimensional domains, we show the existence of solutions to the H-system on an annulus satisfying Neumann boundary conditions.

New minimal surfaces found with spherical curvature lines.

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a p…

In the present work we establish a quantization result for the angular part of the energy of solu- tions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this a…

We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topolog…

This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.

Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.

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

Characterizes conformal classes of tori using differential geometry.

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Study finds non-isotopic transverse tori in Engel manifolds.

New examples show non-rotational annuli in a ball, solving a uniqueness problem.

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…

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

Constructs flows of tori in sphere perturbations for Morse homology.

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…

Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.

Study of critical tori for mean curvature energies in Killing submersions.

We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.

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…

Study tiling spaces over irrational tori using diffeological classification.

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

New minimal tori found in curved spaces.

Classifies mapping tori of specific groups, generalizing known results.

Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite…

Paper explains dynamics of homeomorphisms to mapping tori geometry.

Smooth tori in S^4 are topologically unknotted.

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…

Counts minimal tori in Riemannian manifolds with 6 or more dimensions.

The paper finds non-contractible loops of Legendrian tori from knot families.

The study limits the number of 2-holed tori in knot exteriors.

Engel manifolds show transverse tori can be made to have various formal invariants.

New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.

We show that for $m>n\geq 2$, there are at least two exact isotropic $n$-tori in $\mathbb{C}^m$ which are not Hamiltonian isotopic in $\mathbb{C}^m$, even though they are smoothly isotopic as isotropic $n$-tori. We apply this discovery to obtain more distinct non-exact isotropic tori in $\mathbb{C}^m$.

We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.

We found unique tori with same curvatures using isometric transformations.

The study characterizes subgroups of mapping tori of free groups.

Free maps exist on low-dimensional tori and closed surfaces.

Delaunay tori minimize Willmore energy under isoperimetric constraints.