arXiv research

New non-cobordant hyperbolic manifolds found in certain dimensions.

Let $X$ be a compact quotient of the product of the real Heisenberg group $H_{4m+1}$ of dimension $4m+1$ and the 3-dimensional real Euclidean space $\bR^3$. A left invariant hypercomplex structure on $H_{4m+1}\times \bR^3$ descends onto the compact quotient $X$. The space $X$ is a hyperholomorphic fibration of 4-tori o…

Closed Riemannian manifolds with positive mixed sectional curvature

In a recent article the first three authors proved that in dimension $4m+1$ all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension $4m-1, m\geq2$ admit Sasakian-…

We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class…

The paper constructs multiple manifolds with similar properties.

We classify indefinite simply connected hyper-Kaehler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m,4m). We establish a natural 1-1 correspondence between simply connected hyper-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H…

Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\primeΛand det L as the metric on M varies, where Δdenotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals de…

The geometry arising from Michelson & Strominger's study of N=4B supersymmetric quantum mechanics with superconformal D(2,1;alpha)-symmetry is a hyperKaehler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters alpha are related via changes in potentials for the HKT targe…

Finite volume ends found in quaternionic Kähler manifolds.

Constructs chiral rational homology spheres with hyperbolic groups.

Let M be a positive quaternionic Kaehler manifold of dimension 4m. If the isometry group Isom(M) has rank at least m/2 +3, then M is isometric to HP^m or Gr_2(C^{m+2}). The lower bound for the rank is optimal if m is even.

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature $(2m,2m)$ with $m\ge 3$. Moreover, the geometry of the second factor is encoded i…

Paper finds topological block to certain Lagrangian submanifolds.

In this paper we give a survey of the constructions in math.DG/0510061 of several new invariants for CR and contact manifolds. The latter extend previous constructions of Hirachi and Boutet de Monvel. In addition, we give simple algebro-geometric arguments proving that Hirachi's invariant vanishes on strictly pseudocon…

Constructs Einstein metrics on manifolds with specific orbits.

The paper describes Seiberg-Witten equations in all dimensions.

We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$ is even, and $r \in [0,4] \cup (\frac{4m}ω+4, 2m+4)$ if $m$ is odd, where $ω>1$ is the unique real solution of the equat…

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. Th…

The study finds positive Einstein metrics on complex manifolds and spheres.

We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\script…

Ricci-positive manifolds span the kernel of the $\hat A$-genus in rational Spin bordism.

This paper completes proofs for left orderable slopes of double twist knots.

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits tw…

Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.

The paper studies invariants of almost complex and almost Kähler manifolds, proving properties and relationships between them.

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of π/m. They are invariant under a finite group of screw motions and have no reflection symmetry across a great…

For each nonnegative integer we find an open (4m+9)-dimensional simply-connected manifold admitting complete nonnegatively curved metrics whose souls are non-diffeomorphic, homeomorphic, and have codimension 2. We give a diffeomorphism classification of the pairs (N, soul) when N is a nontrivial complex line bundle ove…

We show that the Atiyah-Patodi-Singer reduced $η$-invariant of the twisted Dirac operator on a closed $4m-1$ dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight $2m$ up to an integral $q$-series. We prove this resu…

We give a complete obstruction to turning an immersion of an m-dimensional manifold M in Euclidean n-space into an embedding when 3n>4m+4. It is a secondary obstruction, and exists only when the primary obstruction, due to Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implie…

Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle …

The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.

A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a p…

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the b…

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKähler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then s…

In [ES2], the first and the third authors introduced new classes in the Johnson cokernels of the mapping class groups of surfaces by a representation theoretic approach based on some previous results for the Johnson cokernels of the automorphism groups of free groups. On the other hand, in [KK1], Kawazumi and the secon…

Study finds new knot distances and chirally cosmetic bands using grid diagrams.

RiskNet predicts penalties in unreliable communication networks using GNNs.

We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, n > 1 with n odd the exterior of the Montesinos knot K(-1/2, m/(2m \…

In this note we answer a question of G. Lecué, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector …

NFTs with diverse rare attributes sell at higher prices.

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, thi…

To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…

Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings. Instead, we consider diagrams enumerated by bridge number, following the lead of S…

The paper provides non-asymptotic Edgeworth expansions for neural network outputs.

For certain metrics, the paper finds that the sixth-order Q-curvature is positive in some dimensions but negative in others.

Aggregate and systemic risk in complex systems are emergent phenomena depending on two properties: the idiosyncratic risks of the elements and the topology of the network of interactions among them. While a significant attention has been given to aggregate risk assessment and risk propagation once the above two propert…