We identify all Anosov representations of compact hyperbolic triangle reflection groups into the higher rank Lie group $\mathrm{SL}(3, \mathbb{R})$. Specifically, we prove that such a representation is Anosov if and only if either it lies in the Hitchin component of the representation space, or it lies in the "Barbot component" and the product of the three generators of the triangle group has distinct real eigenvalues. Unlike representations in the Hitchin component, Anosov representations in the Barbot component have non-convex boundary maps.
In this paper, we proved the existence of n+5/2 prime closed geodesics on a bumpy Finsler sphere (S-n, F) with an odd integer n >= 3 if all geometrically distinct prime closed geodesics have positive Morse indices.
First we construct minimal hypersurfaces M C Rn+1 in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C & times; R, for any strictly minimizing strictly stable cone C in R-n. We show that many of these hypersurfaces are area minimizing. Next, we prove a strong unique continuation result for minimal hypersurfaces V with such a cylindrical tangent cone, stating that if the blowups of V centered at the origin approach C & times; R at infinite order, then V = C & times; R in a neighborhood of the origin. Using this we show that for quadratic cones C = C(S-p & times; S-q), in dimensions n > 8, all O(p + 1) & times; O(q + 1)-invariant minimal hypersurfaces with tangent cone C & times; R at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to C & times; R or has an isolated singularity at the origin.
Let E-tau be the torus with periods 1, tau and delta(0) be the Dirac measure at 0. Consider the following curvature equation (0.1) triangle u + e(u)= 8 pi n delta(0) on E tau, and the integral Lame equation (0.2) y '' = (n(n + 1)& wp;(z | tau) + B)y, where & wp;(z | tau) is the Weierstrass elliptic function. Motivated by our previous works, we conjecture that solvability of (0.1) depends on the geometry of E-tau determined by the multiple Green function G(n) of E-tau. The main purpose of this paper is to confirm our conjecture. Let F-0 be a fundamental domain of Gamma(2) (see (1.4) for the definition). Define LWn = the continuous part of {tau E F-0 | G(n)(z(1), ..., z(n); tau) has a degenerate trivial critical point}. Among other things, we prove the following results. (i) A necessary and sufficient condition for the existence of solutions of (0.1) with tau E F-0, that is, there are n(n+1)/ 2 simply connected open domains Lambda((k)) (j) in F-0 such that (0.1) has an even solution iff tau E U- j,U-k Lambda((k)) (j) and partial derivative Lambda((k))(j)subset of LWn. (ii) If the monodromy matrices S-i, i = 1, 2, of (0.2) are unitarizable, and lambda is an eigenvalue of S-1 or S-2, then lambda is an element of/ {+/- 1}. The is the best possible result for the integral Lame equation (0.2). (iii) For n = 2, 3, 4, we prove the asymptotic behavior of Re tau for those tau where (0.1) has a solution, as tau -> oc. We remark that the paper is a culmination of our previous works over more than ten years.
In arXiv:1911.08213 it was conjectured that the compactly supported cohomology of the m-th restricted contact locus of an isolated hypersurface singularity coincides, up to a shift, with the Floer cohomology of the m-th iterate of the monodromy of the Milnor fiber. In this paper we give an affirmative answer to this conjecture in the case of plane curves.
