Diffusion by optimal transport in Heisenberg groups

We prove that the hypoelliptic diffusion of the Heisenberg group \({\mathbb{H }}_n\) describes, in the space of probability measures over \({\mathbb{H }}_n\) , a curve driven by the gradient flow of the Boltzmann entropy \({{\mathrm{Ent}}}\) , in the sense of optimal transport. We prove that conversely any gradient flow curve of \({{\mathrm{Ent}}}\) satisfy the hypoelliptic heat equation. This occurs in the subRiemannian \({\mathbb{H }}_n\) , which is not a space with a lower Ricci curvature bound in the metric sense of Lott–Villani and Sturm.     

Relaxed Hyperelastic Curves on a Non-Degenerate Surface

We study the variational problem belonging to a relaxed hyperelastic curve for non-null curve on a non-degenerate surface in Minkowski three-space \({E_{1}^{3}}\) . Firstly, we derive the intrinsic equations for a relaxed hyperelastic curve and we give the necessary condition for being relaxed hyperelastic curve of any non-null geodesic on the surface in \({E_{1}^{3}}\) . Then, we examine this formulation on non-null geodesics of pseudo-plane, pseudo-sphere \({S_{1}^{2}(r) }\) , hyperbolic space \({H_{0}^{2}(r)}\) and pseudo-cylinder \({C_{1}^{2}(r)}\) .     

On Maximally Inflected Hyperbolic Curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert’s method we show that for any integers \(d\) and \(r\) such that \(4\le r \le 2d^2-2d\) , there is a non-singular hyperbolic curve of degree \(2d\) in \({\mathbb R}^2\) with exactly \(r\) line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree \(6\) .     

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\) . For \(d\le 7\) , all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\) ; see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\) , see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\) ; see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\) , up to projectivity. For \(d=11,13,17,19\) , this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).     

Distinct Distances on Curves Via Rigidity

It is shown that $N$ points on a real algebraic curve of degree $n$ in ${\mathbb R}^d$ always determine $\gtrsim _{n,d}$ ${N^{1+\frac{1}{4}}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of $N$ points which determine $\lesssim $ ${N}$ distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.     

The Brill–Noether curve and Prym-Tyurin varieties

We prove that the Jacobian of a general curve C of genus $g=2a+1$ , with $a\ge 2$ , can be realized as a Prym-Tyurin variety for the Brill–Noether curve $W^{1}_{a+2}(C)$ . As consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series $g^{a-1}_{3a-2}$ .     

Duality and the topological filtration

Let $(M,J)$ be a Fano manifold which admits a Kähler-Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $g_{KS}$ ). Then we prove that Kähler-Ricci flow on $(M,J)$ converges to $g_{KE}$ (or $g_{KS}$ ) in $C^\infty $ in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to $g_{KE}$ (or $g_{KS}$ ). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.     

Elastic Splines I: Existence

Given interpolation points \(P_1,P_2,\ldots ,P_m\) in the plane, it is known that there does not exist an interpolating curve with minimal bending energy unless the given points lie sequentially along a line. We say that an interpolating curve is admissible if each piece, connecting two consecutive points \(P_i\) and \(P_{i+1}\) , is an s-curve, where an s-curve is a planar curve which first turns monotonically at most \(180^\circ \) in one direction and then turns monotonically at most \(180^\circ \) in the opposite direction. Our main result is that among all admissible interpolating curves there exists a curve with minimal bending energy. We also prove, in a very constructive manner, the existence of an s-curve, with minimal bending energy, that connects two given unit tangent vectors.     

Toric surfaces,K-stability and Calabi flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$ . Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $ \int _{\partial P} u d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$ . Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson’s $M$ -condition holds for $u$ . As an application, we show that if $(X,P)$ is (analytic) relative $K$ -stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.     

A backward \lambda -lemma for the forward heat flow

The inclination or \(\lambda \) -lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold \(M\) provided by the heat flow. The main result is a backward \(\lambda \) -lemma for the heat flow near a hyperbolic fixed point \(x\) . There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our \(\lambda \) -lemma moves the given disk transversal to the unstable manifold backward in time, although there is no backward flow. As a first application we propose a new method to calculate the Conley homotopy index of \(x\) .     

Ricci flow of homogeneous manifolds

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $\mathcal H _{q,n}$ of the variety of $(q+n)$ -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $n$ with a $q$ -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $2$ -parameter subspace of $\mathcal H _{1,3}$ reaching most of $3$ -dimensional geometries, and a $2$ -parameter family in $\mathcal H _{0,n}$ of left-invariant metrics on $n$ -dimensional compact and non-compact semisimple Lie groups.     

Regularity and uniqueness of a class of biharmonic map heat flows

We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\) , that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \) . We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\) , then we establish the local existence of heat flow of biharmonic maps \(u\) , with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).     

Geometric flows and differential Harnack estimates for heat equations with potentials

Let $M$ be a closed Riemannian manifold with a Riemannian metric $g_{ij}(t)$ evolving by a geometric flow $\partial _{t}g_{ij} = -2{S}_{ij}$ , where $S_{ij}(t)$ is a symmetric two-tensor on $(M, g(t))$ . Suppose that $S_{ij}$ satisfies the tensor inequality $2{\mathcal H}(S, X)+{\mathcal E}(S,X) \ge 0$ for all vector fields $X$ on $M$ , where ${\mathcal H}(S, X)$ and ${\mathcal E}(S,X)$ are introduced in Definition 1 below. Then, we shall prove differential Harnack estimates for positive solutions to time-dependent forward heat equations with potentials. In the case where $S_{ij} = R_{ij}$ , the Ricci tensor of $M$ , our results correspond to the results proved by Cao and Hamilton (Geom Funct Anal 19:983–989, 2009). Moreover, in the case where the Ricci flow coupled with harmonic map heat flow introduced by Müller (Ann Sci Ec Norm Super 45(4):101–142, 2012), our results derive new differential Harnack estimates. We shall also find new entropies which are monotone under the above geometric flow.     

On the roots of the equation \zeta (s)=a

Given any complex number $a$ , we prove that there are infinitely many simple roots of the equation $\zeta (s)=a$ with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve $t\mapsto \zeta ({1\over 2}+it)$ . Moreover, we show that the curve $\mathbb {R}\ni t\mapsto (\zeta ({1\over 2}+it),\zeta '({1\over 2}+it))$ is not dense in $\mathbb {C}^2$ .     

Invariant theory for the elliptic normal quintic, I. Twists of X(5)

A genus one curve of degree 5 is defined by the $4 \times 4$ Pfaffians of a $5 \times 5$ alternating matrix of linear forms on $\mathbb{P }^4$ . We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants and to extend our method for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the $5$ -Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.     

On Sets Defining Few Ordinary Lines

Let $P$ P be a set of $n$ n points in the plane, not all on a line. We show that if $n$ n is large then there are at least $n/2$ n / 2 ordinary lines, that is to say lines passing through exactly two points of $P$ P . This confirms, for large $n$ n , a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $n-C$ n - C ordinary lines for some absolute constant $C$ C . We also solve, for large $n$ n , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $P$ P . Underlying these results is a structure theorem which states that if $P$ P has at most $Kn$ K n ordinary lines then all but O(K) points of $P$ P lie on a cubic curve, if $n$ n is sufficiently large depending on $K$ K .     

Approximations of periodic functions to \mathbb R ^n by curvatures of closed curves

We show that for any $n$ real periodic functions $f_1,\ldots , f_n$ with the same period, such that $f_i>0$ for $i<n$ , and a real number $\varepsilon >0$ , there is a closed curve in $\mathbb R ^{n+1}$ with curvatures $\kappa _1, \ldots , \kappa _n$ such that $\left| \kappa _{i(t)}-f_{i(t)}\right|<\varepsilon $ for all $i$ and $t$ . This does not hold for parametric families of closed curves in $\mathbb R ^{n+1}$ .     

A note on Griffiths infinitesimal invariant for curves

Given a generic curve of genus $g\ge 4$ and a smooth point $L\in W_{g-1}^{1}(C)$ , whose linear system is base-point free, we consider the Abel–Jacobi normal function associated with $L^{\otimes 2}\otimes \omega _{C}^{-1}$ , when $(C,L)$ varies in moduli. We prove that its infinitesimal invariant reconstructs the couple $(C,L)$ . When $g=4$ , we obtain the generic Torelli theorem proved by Griffiths.     

Tracing Compressed Curves in Triangulated Surfaces

A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. A normal curve is a finite set of disjoint normal paths and normal cycles. We describe an algorithm to “trace” a normal curve in $O(\min \{ X, n^2\log X \})$ O ( min { X , n 2 log X } ) time, where $n$ n is the complexity of the surface triangulation and $X$ X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in $n$ n . Our tracing algorithm computes a new cellular decomposition of the surface with complexity $O(n)$ O ( n ) ; the traced curve appears in the 1-skeleton of the new decomposition as a set of simple disjoint paths and cycles. We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer et al. (Proceedings of 8th International Conference Computing and Combinatorics. Lecture Notes in Computer Science, vol. 2387, pp. 370–380. Springer, Berlin 2002; Proceedings of 20th Canadian Conference on Computational Geometry, pp. 111–114, 2008) and by Agol et al. (Trans Am Math Soc 358(9): 3821–3850, 2006).