For a local non-toric Calabi–Yau manifold which arises as a smoothing of a toric Gorenstein singularity, this paper derives the open Gromov–Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its Strominger-YauZaslow (SYZ) mirror. Moreover, it proves that the SYZ mirrors and disk potentials vary smoothly under conifold transitions, giving a global picture of SYZ mirror symmetry

We show that the recently proposed weak gravity conjecture\cite{AMNV0601} can be extended to a class of scalar field theories. Taking gravity into account, we find an upper bound on the gravity interaction strength, expressed in terms of scalar coupling parameters. This conjecture is supported by some two-dimensional models and noncommutative field theories.

Abstract We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura vari- eties. As a crucial ingredient, we formulate a conjectural arithmetic funda- mental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

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We consider compact submanifolds of dimension n  2 in Rn+k, with nonzero mean curvature vector everywhere, and such that the full norm of the second fundamental form is bounded by a fixed multiple (depending on n) of the length of the mean curvature vector at every point. We prove that the mean curvature flow deforms such a submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in some (n + 1)- dimensional affine subspace of Rn+k.