Related papers: Geometrically incompressible non-orientable closed…

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously…

In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using…

We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which…

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten…

We enumerate cubic (3-regular) unicellular maps on closed surfaces up to all homeomorphisms. Using the orbifold approach, we reduce the unsensed enumeration to explicit counts of quotient maps and rooted cubic/precubic maps on simpler…

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge of the…

A celebrated result of Mirzakhani states that, if $(S,m)$ is a finite area \emph{orientable} hyperbolic surface, then the number of simple closed geodesics of length less than $L$ on $(S,m)$ is asymptotically equivalent to a positive…

Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some…

We present a second order numerical scheme to compute capillary bridges between arbitrary solids by minimizing the total energy of all interfaces. From a theoretical point of view, this approach can be interpreted as the computation of…

Let $\Delta$ be a $d$-dimensional normal pseudomanifold, $d \ge 3.$ A relative lower bound for the number of edges in $\Delta$ is that $g_2$ of $\Delta$ is at least $g_2$ of the link of any vertex. When this inequality is sharp $\Delta$ has…

A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit…

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains…

Let $\Sigma$ be a surface with boundary $b(\Sigma)$, $\mathcal{L}$ be a collection of $k$ disjoint $b(\Sigma)$-paths in $\Sigma$, and $P$ be a non-separating $b(\Sigma)$-path in $\Sigma$. We prove that there is a homeomorphism $\phi: \Sigma…

Let $\F$ be a compact surface and let $I$ be the unit interval. This paper gives a standard form for all 2-sided incompressible surfaces in the 3-manifold $\F \times I$. Since $\F \times I$ is a handlebody when $\F$ has boundary, this…

We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a…

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of…

Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted, so that the crossing number of $\mathcal{K}$ is zero. We determine the…

For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal…

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a…

Let $f$ be a Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ and $\chi$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa\geq 10$. We prove a subconvexity bound $$…