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Robert Gulliver

Publications and source records attributed to Robert Gulliver.

7 recordsLinked to original sources
arXiv2017

Density of a minimal submanifold and total curvature of its boundary

Given a piecewise smooth submanifold $Γ^{n-1} \subset \R^m$ and $p \in \R^m$, we define the {\em vision angle} $Π_p(Γ)$ to be the $(n-1)$-dimensional volume of the radial projection of $Γ$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $Σ\subset \R^m$ with boundary $Γ$, then we show the density of $Σ$ at $p$ is $\leq$ the density at its vertex $p$ of the cone over $Γ$. It follows that if $Π_p(Γ)$ is less than twice the volume of $S^{n-1}$, for all $p \in Γ$, then $Σ$ is an embedded submanifold. As a consequence, we prove that given two $n$-planes $R^n_1, R^n_2$ in $\R^m$ and two compact convex hypersurfaces $Γ_i$ of $R^n_i, i=1,2$, a nonflat minimal submanifold spanned by $Γ:=Γ_1\cupΓ_2$ is embedded.

math.DG↗
arXiv2013

Branch points of area-minimizing projective planes

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those mappings which are not homotopic to a constant mapping, is an immersion. That is, $f$ is free of branch points. As a major step toward treating minimal surfaces of the type of the projective plane, we extend the fundamental theorem of branched immersions to the nonorientable case. We also resolve a question on the directions of branch lines posed by Courant in 1950.

math.DG↗
arXiv2011

Total Curvature of Graphs after Milnor and Euler

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.

math.DG↗
arXiv2009

Regularity of soap film-like surfaces spanning graphs in a Riemannian manifold

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-κ^2$. Using the cone total curvature $TC(Γ)$ of a graph $Γ$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $Σ$ spanning a graph $Γ\subset M$ is less than or equal to $\frac{1}{2π}\{TC(Γ) - κ^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\times$}Γ)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when $n=3$, this density estimate implies that if \begin{eqnarray*} TC(Γ) < 3.649π+ κ^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}Γ}), \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,δ)$-minimizing set $Σ$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $π/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.

math.DG↗
arXiv2009

Examples of hypersurfaces flowing by curvature in a Riemannian manifold

This paper gives some examples of hypersurfaces $ϕ_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1 to $n$, and include cases which exist for infinite time. Convergence to a point was studied by Andrews, and only occurs in finite time. For dimension $n=2,$ the destiny of any harmonic mean curvature flow is strongly influenced by the genus of the surface $M^2$.

math.DG↗
arXiv2008

Total curvature and isotopy of graphs in $R^3$

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs $Γ$, what limitations on the isotopy class of $Γ$ are implied by a bound on total curvature? What does ``total curvature" mean for a graph? We define a natural notion of net total curvature of a graph $Γ$ in $R^3$, and prove that if $Γ$ is homeomorphic to the $θ$-graph, then the net total curvature of $Γ$ \geq 3π$; and if it is $< 4π$, then $Γ$ is isotopic in $R^3$ to a planar $θ$-graph. Further, the net total curvature $= 3π$ only when $Γ$ is a convex plane curve plus a chord. We begin our discussion with piecewise smooth graphs, and extend all these results to continuous graphs in the final section. In particular, we show that continuous graphs of finite total curvature are isotopic to polygonal graphs.

math.DG↗