arXiv research

Study bounds the Morse index of a special torus to 1.

The Clifford torus minimizes Willmore energy closely for small perturbations.

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

The paper studies Willmore surfaces in 4D conformal manifolds and finds the Clifford torus is strictly Willmore-stable.

We show that Bonnesen's isoperimetic defect has a systolic analog for Loewner's torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal factor of the metric g with respect to the flat metric of unit area in the conformal class of g.

The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence …

Optimizes the conformal capacity of linked curves in $S^3$.

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

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such t…

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constan…

Study shows that deformed Liouville metrics on tori remain Liouville.

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…

For every two-dimensional torus $T^2$ and every $k\in \mathbb{N}$, $k\ge 3$, we construct a conformal Willmore immersion $f:T^2\to \mathbb{R}^4$ with exactly one point of density $k$ and Willmore energy $4πk$. Moreover, we show that the energy value $8π$ cannot be attained by such an immersion. Additionally, we charact…

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

We show how to assign to any immersed torus in $\R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the toru…

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface Σ, such that f can be reconstructed in terms of algebraic data on Σ. Particularly interesting exam…

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…

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisma…

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent …

Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.

We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed.…

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minim…

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…

We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not c…

Study of tori of revolution under Willmore flow converges to Clifford Torus.

We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a positive real number. Moreover, we prove that any toric locally conformally Kähler me…

It is known that automorphism group $G$ of a compact homogeneous locally conformally Kähler manifold $M=G/H$ has at least a 1-dimensional center. We prove that the center of $G$ is at most 2-dimensional, and that if its dimension is 2, then $M$ is Vaisman and isometric to a mapping torus of an isometry of a homogeneous…

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 $H\neq 0$. Our main res…

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…

Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisa…

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group …

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …

Ejiri's torus in $S^5$ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a special example. I…

In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schrödinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the potential of the Schrödinger operators, which has a natural geometrical meaning. We …

We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this r…

In this paper, we will study the existence problem of minmax minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of minmax minimal torus in Theorem 5.1. Firstly we prove a strong uniformization result(Proposition 3.1) using method of [1]. Then we use…

For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Mod…

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformall…

We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We pr…

We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar …

Study shows tori metrics converging to flat under specific conditions.

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…

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…

We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $\varphi$ such that $d \varphi = - θ\wedge \varphi$ for a closed non-vanishing 1-form $θ$. Moreover, we show that if $(…

Derives a formula for the second variation of the Laplace eigenvalue functional on manifolds.