Let X be a smooth affine surface, X ! G2 m be a finite morphism. We study the affine curves on X, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on X over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of f(u, v, y) = 0, with u, v, y over a function field, u, v S-units. It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi [12]; however, the algebraic context was left open and seems not to fall in the existing techniques.

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359–409 (2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li–Yau inequality by the global gradient estimate, we can get similar results.

For almost all Riemannian metrics (in the C1 Baire sense) on a closed manifold M^{n+1}, 3 \leq n + 1 \leq 7, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of a result by Irie and the first two authors, that established density of minimal hypersurfaces for generic metrics. The main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors .

Let M be a closed orientable 3-manifold admitting an H2 × R or gSL2(R) geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to S1, according as the center of 1(M) is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings SF(M) for compact orientable aspherical 3-manifolds. We also prove that when M has an H2×R or gSL2(R) geometry and the base orbifold has underlying manifold the 2sphere with three cone points, the inclusion Isom(M) ! Diff(M) is a homotopy equivalence.

The paper deals with a variational approach of the subRiemannian geometry from the point of view of Hamilton-Jacobi and Hamiltonian formalism. We present a discussion of geodesics from the point of view of both formalisms, and prove that the normal geodesics are locally length-minimizing horizontal curves.