Major accomplishments: 1) Heinz-type inequality for graph submanifolds in any codimension. If Γf:={(x,f(x)):x∈ M} is a graph submanifold with parallel mean curvature H of the product M^m × N^n of two Riemannian manifolds, f:(M,g)→(N,h) smooth map, then |H|≤ h(M)/m where h(M) is the Cheeger constant. So, if h(M)=0 then Γf is minimal. This is the case when M is strongly diffeomorphic to a Riemannian manifold with non-negative Ricci tensor[00,1,4]. In case of the hyperbolic space M=B^m (Poincaré disk), h(M)=(m−1), for each c∈[1−m, m−1], I construct a smooth function f(x)=F(r(x)), which graph has CMC |H|=|c|/m, where r(x)= Poincaré distance to 0, [0,1,2]. 2) Mean curvature flow of spacelike graphs (with G.Li [3]). We prove the mean curvature flow of a spacelike graph in (Σ1×Σ1,g1×−g2 ) ofa map f:Σ1→Σ2 remains spacelike, exists for all time, and converges to a slice at infinity as long as (Σ1,g1) is closed with Ricci_1>0, (Σ2,g2) is complete with bounded curvatures and sectional curvature K2≤K1. If K1>0 -no need K2≤K 1, or if Ricci_1>0 and K2≤−c, c>0 constant, f:Σ1→Σ2 is trivially homotopic provided f^∗g2<ρ g1 , where ρ = min_{Σ1} K1/ sup_{Σ2}K2^+ ∈ (0,+∞]. This largely extends some known results. 3) Families of non-tiling domains satisfying Pólya’s conjencture (with P.Freitas [7]). In 1945 Pólya conjectured that the Dirichlet and Neumann eigenvalues of an Euclidean domain Ω, satisfies µ_k ≤ CW(Ω)k^{2/n} ≤ λ_k , ∀k, where CW(Ω)=4π^2/(ω_n|Ω|)^{2/n} is the Weyl constant of Ω, proving to be true for domains that tile R^n. Since then, few other examples were known to saisfy the conjecture. In 2023 we have shown the existence of a large class of non tiling domains that satisfy the conjecture, as for example, sufficiently small sectors of domains of revolution or of geodesic disks of space forms, or solid cylinders and cylindrical surfaces with sufficiently small height. Such domains satisfy two conditions: 1) an inverted Pólya argument holds -for each p∈ N large enough Ω can be partitioned into p-tiles; 2) Ω satisfies Pólya conjecture eventually. Note: 2) holds if Ω satisfies the non-blocking and non-periodicity condition of the geodesic billiards. In particular, for cylinders Ω×[0,l], with Ω⊂ R^{n−1} , the conjecture holds for l small enough. For any height l>0 we improve the Li-Yau-Berezin constant β_n=n/(n+2) on a Pólya type inequality to a larger one. 4) Pólya Conjecture on the Laplacian eigenvalues of a hemispheres (with P.Freitas & J.Mao [8]). In 1980 Bérard & Besson proved that the 2-dimensional hemisphere S_+^2 satisfies Pólya conjecture. If n≥3 we proved the Neumann eigenvalues of S_+^n satisfy Pólya conjecture but not the Dirichlet ones, not even eventually. On the other hand a Pólya type inequality holds with a correction constant second term, that corresponds to an arithmetic and geometric means type inequality. 5) Spherical n-Lunes: Billiards and Eigenvalues (with P.Freitas [9]). We classify the billiards of any lune of S^n , and show the geodesic orbits of the dihedral lunes are all periodic and each one corresponds to an exact origami of a geodesic S^1 of S^n folded into the lune at an explicit number of boundary hits. We prove the Dirichlet spectrum of the dihedral and irrational lunes satisfy Pólya’s conjecture eventually. On dihedral lunes, we estimate bounds on the opening angle θ such that Pólya conjecture holds when θ ≤ θ1(n) and does not if θ ≥ θ2(n), and derive a sharp generalized two-term Weyl asymptotics with positive terms. Selected Works: [00] Graphs with Parallel mean curvature and a variational problem in conformal geometry, Ph.D.Thesis, December 1987, Universiy of Warwick [0] Grassmannian manifolds as subsets of Euclidean spaces, (with A.Machado) Proc. 5th Int. Colloq., Santiago de Compostela, Spain, 1984, Research Notes in Mathematics, 131, 85-102, Pitman Adv. Pub.Prog., Boston-London-Melbourne (1985). (arxiv version: arXiv:2101.09731) (24 cit.) [1] Graphs with parallel mean curvature. Proceedings of the American Mathematical Society 107 (2) (1989), 449-449. 10.2307/2047835. (64 cit.) [2] Spacelike graphs with parallel mean curvature. Bulletin of the Belgian Mathematical Society -Simon Stevin 15(1) (2008): 65-76. (25 cit.) [3] Mean curvature flow of spacelike graphs, (with G.Li). Mathematische Zeitschrift 269, 3-4(2011), 697-719 (35 cit.) [4] Bernstein-Heinz-Chern results in calibrated manifolds, (with G.Li). Revista Matematica Iberoamericana 26(2) (2010), 651-692. [5] The isoperimetric problem in higher codimension, (with F.Morgan). Manuscripta Mathematica 142 (3-4) (2013), 369-382. 10.1007/s00229-012-0604-8. [6] Spherical symerization and the first eigenvalue of geodesic disks on manifolds, (with P.Freitas and J.Mao). Calculus of Variations and Partial Differential Equations, 51(2014), 701-724. 10.1007/s00526-013-0692-7. (22 citations) [7] Families of non-tiling domains satisfying Pólya's conjecture (with P.Freitas). Journal of Mathematical Physics 64, 121503 (2023), 10.1063/5.0161050 (10 citations) [8] Pólya type inequalities on spheres and hemispheres (with P.Freitas and J.Mao) Annales l’Institut Fourier 75 (2025) no. 3, 979-1051. [9] Spherical n-Lunes: Billiards and Eigenvalues (with P.Freitas). Preprint 2025. Anouncements of some results were given in my talk "New examples of Rienmannian domains satisfying Po'lya conjecture on Laplacian eigenvalues". A conference in honours of Marco Rigoli's 70th Birthday, Università degli Studi di Milano, Italy, 16-20 June 2025.