Any component of moduli of polarized hyperkähler manifolds is dense in its deformation space

Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v   in H²(M)$H^2(M)$ defines a divisor _Dv$D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.     

The inverse problem on subset sums

For a set A$A$, let P(A)$P\left(A\right)$ be the set of all finite subset sums of A$A$. We prove that if a sequence B={_b1<_b2<?}$B=\{b_1<b_2<?\}$ of integers satisfies _b1≥11,_b2≥3_b1+5,_b3≥3_b2+3$b_1\ge 11,b_2\ge 3b_1+5,b_3\ge 3b_2+3$ and _bn+1>3_bn−_bn−2$b_{n+1}>3b_n-b_{n-2}$  (n≥3)$(n\ge 3)$, then there exists a sequence of positive integers A={_a1<_a2<?}$A=\{a_1<a_2<?\}$ such that P(A)=N?B$P\left(A\right)=\mathbb{N}?B$. These lower bounds are optimal in a sense. We pose a problem for further research.     

Symplectic Calabi–Yau 6-manifolds

In this article, we construct simply connected symplectic Calabi–Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T⁴${\mathbb{T}}^4$. Using our method, we also construct symplectic non-Kähler Calabi–Yau 6-manifolds with fundamental group Z$\mathbb{Z}$. This paper also produces the first examples of simply connected and non-simply connected symplectic Calabi–Yau 6-manifolds with fundamental groups _Zp×_Zq${\mathbb{Z}}_p\times {\mathbb{Z}}_q$, and Z×_Zq$\mathbb{Z}\times {\mathbb{Z}}_q$ for any p≥1$p\ge 1$ and q≥2$q\ge 2$via co-isotropic Luttinger surgery.     

Distribution functions for subsequences of the van der Corput sequence

For an integer b>1$b>1$ let _(?b(n))n≥0${\left({?}_b\left(n\right)\right)}_{n\ge 0}$ denote the base b$b$ van der Corput sequence in [0,1)$[0,1)$. Answering a question of O. Strauch, C. Aisleitner and M. Hofer showed that the distribution function of _{(?b(n),?b(n+1),…,?b(n+s−1))n≥0}${({?}_b\left(n\right),{?}_b(n+1),\dots ,{?}_b(n+s-1))}_{n\ge 0}$ on [0,1)^s${[0,1)}^s$ exists and is a copula. In this note we show that this phenomenon extends to a broad class of subsequences of the van der Corput sequences.     

Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and _Cb(X)$C_b(X)$ be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T   from _Cb(X)$C_b(X)$ to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=_T1+_T2+_T3+_T4$T=T_1+T_2+T_3+T_4$, where _Tk:_Cb(X)→E$T_k:C_b(X)\to E$(k=1,2,3,4)$(k=1,2,3,4)$ are weakly compact operators, and _T1$T_1$ is tight, _T2$T_2$ is purely τ  -additive, _T3$T_3$ is purely σ  -additive and _T4$T_4$ is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.     

Some applications of Gabaiʼs internal hierarchy

Any Haken 3-manifold (possibly with boundary consisting of tori) can be transformed into a surface×S¹$\text{surface}\times S^1$ by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few technical details in the proof. In the second half of this paper, some applications of Gabai?s theorem to Heegaard Floer homology are given. We refine the known results about the Thurson norm and fibrations. We also give some classification results for Floer simple knots in manifolds with positive _b1$b_1$.     

Lower bounds on the modified K-energy and complex deformations

Let (X,L)$(X,L)$ be a polarized Kähler manifold that admits an extremal metric in _c1(L)$c_1(L)$. We show that on a nearby polarized deformation (X^′,L^′)$(X^{\prime },L^{\prime })$ that preserves the symmetry induced by the extremal vector field of (X,L)$(X,L)$, the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti ,  and  to extremal metrics. Our proof also extends a convexity inequality on the space of Kähler potentials due to X.X. Chen [7] to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of CP¹×CP¹$\mathbb{C}{\mathbb{P}}^1\times \mathbb{C}{\mathbb{P}}^1$ that carry no extremal metric but with modified K-energy bounded from below.     

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X   such that _b2(X)?1$b_2(X)?1$ we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G  , with prescribed values on certain loops. If _b2(X)=1$b_2(X)=1$ we assume G=SO(3)$G=\mathrm{SO}(3)$ and that the cup product on H¹(X;Q)$H^1(X;\mathbb{Q})$ is non-zero.     

A note on the growth of Betti numbers and ranks of 3-manifold groups

Let N   be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. We show that it follows from the work of Agol, Kahn–Markovic and Przytycki–Wise that _π1(N)${\pi }_1(N)$ admits a cofinal filtration with ‘fast’ growth of Betti numbers as well as a cofinal filtration of _π1(N)${\pi }_1(N)$ with ‘slow’ growth of ranks.     

The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions

In this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (_Eλ,μ)$\left(E_{\lambda ,\mu }\right)$ with sign-changing weight function. With the help of the Nehari manifold, we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ)$(\lambda ,\mu )$ belongs to a certain subset of R²${\mathbb{R}}^2$.     

Localization of favorite points for diffusion in a random environment

For a diffusion _Xt$X_t$ in a one-dimensional Wiener medium W$W$, it is known that there is a certain process _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$ that depends only on the environment, so that _Xt−_blogt(W)$X_t-b_{logt}\left(W\right)$ converges in distribution as t→∞$t\to \infty$. The paths of b$b$ are step functions. Denote by _FX(t)$F_X\left(t\right)$ the point with the most local time for the diffusion at time t$t$. We prove that, modulo a relatively small time change, the paths of the processes _(br(W))r≥0${\left(b_r\left(W\right)\right)}_{r\ge 0}$, _{(FX(er))r≥0}${\left(F_X\left(e^r\right)\right)}_{r\ge 0}$ are close after some large r$r$.     

Nowhere-zero 3-flows and modulo k-orientations

The main theorem of this paper provides partial results on some major open problems in graph theory, such as Tutte?s 3-flow conjecture (from the 1970s) that every 4-edge connected graph admits a nowhere-zero 3-flow, the conjecture of Jaeger, Linial, Payan and Tarsi (1992) that every 5-edge-connected graph is _Z3$Z_3$-connected, Jaeger?s circular flow conjecture (1984) that for every odd natural number k?3$k?3$, every (2k−2)$(2k-2)$-edge-connected graph has a modulo k  -orientation, etc. It was proved recently by Thomassen that, for every odd number k?3$k?3$, every (2k²+k)$(2k^2+k)$-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G   is _Z3$Z_3$-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassen?s method is refined to prove the following: For every odd number  k?3$k?3$, every  (3k−3)$(3k-3)$-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph is  _Z3$Z_3$-connected and admits therefore a nowhere-zero 3-flow. Note that it was proved by Kochol (2001) that it suffices to prove the 3-flow conjecture for 5-edge-connected graphs.     

Local path fitting: A new approach to variational integrators

A new approach for constructing variational integrators is presented. In the general case, the estimation of the action integral in a time interval [_tk,_tk+1]$[t_k,t_{k+1}]$ is used to construct a symplectic map (_qk,_qk+1)→(_qk+1,_qk+2)$(q_k,q_{k+1})\to (q_{k+1},q_{k+2})$. The basic idea, is that only the partial derivatives of the estimated action integral of the Lagrangian are needed in the general theory. The analytic calculation of these derivatives, gives rise to a new integral that depends on the Euler–Lagrange vector itself (which in the continuous and exact case vanishes) and not on the Lagrangian. Since this new integral can only be computed through a numerical method based on some internal grid points, we can locally fit the exact curve by demanding the Euler–Lagrange vector to vanish at these grid points. Thus, the integral vanishes, and the process dramatically simplifies the calculation of high order approximations. The new technique is tested in high order solutions of the two-body problem with high eccentricity (up to 0.99) and of the outer planets of the solar system.     

A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.