Bruce Solomon
We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent indicatrix.
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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-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.
We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $π$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.
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 case. This surface satisfies good properties and is theoretically distinct from traditional examples.
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.
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 surfaces with good property in a 3-torus distinct from classical examples.
Orthogonal complex structures on flat tori
本文分类了平坦环面上的正交复结构，等价于分类具有平坦Levi-Civita联络的紧致Hermitian流形，推广了2017年复三维情形的结果。
Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients).
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 study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that the flat n-tori are focally rigid, in the sense that if two flat tori are focally equivalent, then th…
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.
We describe the space of isometric immersions from the Lorentz plane $L^2$ into the 3-dimensional anti-de Sitter space, and solve several open problems of this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immersed in the anti de Sitter space in terms of pairs of closed curves with wave front singularities in the hyperbolic plane satisfying some compatibility conditions.
On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori
本文证明了平坦 $n$-环面 Willmore 能量的尖锐下界，并确认 Clifford $n$-环面为唯一极小化子，从而证实 Chen 猜想；但 $n\geq3$ 时该猜想对一般浸入环面不成立。
This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $λ_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many…
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.
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 subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show t…
Minimal isometric immersions of flat n-tori into spheres
该研究修正了Bryant关于平坦环面极小等距浸入的有理判据，证明n≥3时该判据仅为充分条件，推导了无理性度上界，构造了非齐次例子并建立了变形定理。
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 riemannian torus of any dimension is secure if and only if it is flat. We prove this for two-dimensional tori.
In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.
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.
Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not parallel. We prove that if an oriented closed manifold has a non-zero Euler characteristic $ξ$ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than $(ξ^2)/4$. We also construct examples of skew odd-dimensional spheres and skew two-dimensional tori.
We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singular homology of the loop space. In an appendix we perturb the equations in order to eliminate degeneracies and to get to a situation with nondegenerate critical points only. The (co)homology groups turn out to be invariant under the perturbation.
Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichmüller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of the Thurston, Teichmüller, and Weil-Petersson metrics. We show the Teichmüller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ as a symmetr…
We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $γ$ in such a torus is said to be homologically maximizing if one (hence every) lift of $γ$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics. This yields the Lipschitz continuity of the time separation of the universal cover on strict sub-cones of the cone of future pointing vectors. Then we introduce the stable time separation $\mathfrak{l}$. As an application we prove relations between the concavity properties of $…
Flat minimal tori and Lu's second-gap conjecture
构造了任意余维数≥3的闭嵌入极小曲面，证明陆氏第二间隙猜想不成立：线性满嵌入平坦极小环面的S+λ₂常数值在(2,3)中稠密。
A fibration of ${\mathbb R}^n$ by oriented copies of ${\mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${\mathbb R}^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of ${\mathbb R}^n$ by sk…
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 the momenta.
The isospectral problem for flat tori from three perspectives
本文从分析、几何和数论三个视角研究平坦环面的等谱非等距问题，回顾Milnor的16维例子，并介绍Schiemann在1990年代对最低存在维数问题的解答及若干未解决问题。
A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold $(M,g)$ determines the Riemannian metric $g$ up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given Riemannian manifold $(Σ,h)$ to $M$ determines the Riemannian metric on the target space. We consider this question in the case of harmonic maps between flat tori. In particular, we show that the two isospectral, non-isometric $16$-dimensional flat tori found by Milnor cannot be distinguished by the energy spectrum of harmonic maps from $d$-dimensional flat tori f…
Consider a compact locally symmetric space $M$ of rank $r$, with fundamental group $Γ$. The von Neumann algebra $\vn(Γ)$ is the convolution algebra of functions $f\in\ell_2(Γ)$ which act by left convolution on $\ell_2(Γ)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $Γ_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $Γ$. Then $\vn(Γ_0)$ is a maximal abelian $\star$-subalgebra of $\vn(Γ)$ and its unitary normalizer is as small as possible. If $M$ has constant negative curvature then the Pukánszky invariant of $\vn(Γ_0)$ is $\infty$.
In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} Δu$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows that unlike harmonic map heat flow, this flow preserves diffeomorphisms. A change of variables for Du then allows us to establish a $C^α$ estimate for the coefficient of the tension field, and thus (thanks to the quasilinear structure and the Schauder estimates) we get full regulari…
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 the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori.
If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.
Let M be a real 2m-torus equipped with a translation-invariant metric h and a translation-invariant symplectic form w; the latter we interpret as a magnetic field on M. The Hamiltonian flow of half the norm-squared function induced by h on T^*M (the "kinetic energy") with respect to the twisted symplectic form w_{T^*M}+ π^*w describes the trajectories of a particle moving on M under the influence of the magnetic field w. If [w] is an integral cohomology class, then we can study the geometric quantization of the symplectic manifold (T^*M,w_{T^*M}+π^*w) with the kinetic energy Hamiltonian. We sa…
The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.
Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L} \rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{L}…
Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes to infinity, and characterise the flat tori using the tools of homogenisation our conclusion being that "Macroscopically, one can hear the shape of a flat torus".
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 flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.
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 considered. We study the determinant of Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeros and derive formulas for variations of this functional w. r. t. natural coordinates on this moduli space. A new proof of …
Ergodic and Entropic Behavior of the Harmonic Map Heat Flow to the Moduli Space of Flat Tori
本文研究调和映射热流在平坦环面模空间中的遍历行为与熵性质，证明流对能量泛函的稳定性及测度收敛性，同时建立几何流与信息论收敛的联系。
Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds
本文提出到平坦环面的非线性调和分裂映射，证明其存在性与 Ricci 及积分标量曲率界下的 Gromov-Hausdorff 靠近性等价，并结合 Stern 不等式建立了平坦 3-环面的新稳定性定理。
This note presents some of the results obtained in arXiv:2207.05410 and it has beenthe object of a talk of the second author during the Journées "Équations auxDérivées Partielles" (Obernai, june 2022). We study properties of geodesics that are orthogonal to two convex subsets of the flat torus. We discuss meromorphic properties of a geometric Epstein zeta function associated to the set of lengths of such orthogeodesics. We also define the associated length distribution and discuss singularities of its Fourier transform. Our analysis relies on a fine study of the dynamical correlation function …
This paper focuses on the development of harmonic and Clifford analysis techniques in the context of some conformally flat manifolds that arise from factoring out a simply-connected domain from $R^n$ by special arithmetic subgroups of the conformal group. Our discussion encompasses in particular the Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori and polycylinders. This paper provides a continuation as well as an extension of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, …
Warped Tori with Almost Non-Negative Scalar Curvature
本文研究了3-环面上满足几乎非负数量曲率及一致几何界条件的扭曲积度量序列，证明了其存在子序列在Gromov-Hausdorff和Sormani-Wenger Intrinsic Flat意义下收敛至平坦3-环面。
Solvability of a class of fully nonlinear elliptic equations on tori
该研究聚焦平坦环面上一类完全非线性椭圆方程的可解性，这类方程源自环面丛相关Calabi-Yau型问题的研究，为相关几何分析问题提供支撑。
Length orthospectrum of convex bodies on flat tori
类比负曲率流形Pollicott-Ruelle共振研究，定义适配平坦环面平移不变Finsler度量测地向量场的各向异性Sobolev空间，研究凸体正交测地线性质，定义几何Epstein函数并证其亚纯延拓、算留数，证正交测地线长度与磁Laplacian谱的Poisson型求和公式。
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