Let M be a compact Riemannian manifold, π : \tilde{M} → M be the universal covering and ω be a smooth 2-form on M with π^∗ω cohomologous to zero. Suppose the fundamental group π_1(M) satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth 1-form η on \tilde{M} of linear (resp. bounded) growth such that π^∗ω = dη. As applications, we prove that on a compact Kahler manifold (M,ω) with π^∗ω cohomologous to zero, if π_1(M) is CAT(0) or automatic (resp. hyperbolic), then M is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic (−1)^{dim_RM/2χ(M) ≥ 0 (resp. >0).

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For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds with b+ = 1, as- suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in [31].

In this paper, we prove a scalar curvature rigidity result for geodesic balls in Sn. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in [7].

By using variational method and under generic condition, we show that Arnold diffusion exists in a priori hyperbolic and timeperiodic Hamiltonian systems with multiple degrees of freedom.