arXiv subjects

Adrian Butscher

Publications and source records attributed to Adrian Butscher.

26 records · Page 2Linked to original sources

arXiv2007

Equivariant Gluing Constructions of Contact Stationary Legendrian Submanifolds of the (2n+1)-Sphere

A contact stationary Legendrian submanifold of $S^{2n+1}$ is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial $n$-sphere $S_0$. This paper develops a method for constructing contact stationary (but not minimal) Legendrian submanifolds of $S^{2n+1}$ by gluing together configurations of sufficiently many $U(n+1)$-rotated copies of $S_0$ at isolated points of suitably transverse intersection. The resulting submanifolds are very symmetric; are geometrically akin to a `necklace' of copies of $S_0$ attached to each other by narrow necks and winding a large number of times around $S^{2n + 1}$ before closing up on itself; and are topologically equivalent to $S^1 \times S^{n-1}$. Moreover, they represent wholly new examples of contact stationary embedded submanifolds of $S^{2n + 1}$ and thus give rise to wholly new examples of embedded Hamiltonian stationary cones in $C^{n+1}$.

math.DG↗

arXiv2007

Gluing Constructions Amongst Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere

Four constructions of constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere are given, which should be considered analogues of `classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like `arms'. This last result extends Butscher and Pacard's doubling construction for generalized Clifford tori of small mean curvature.

math.DG↗

arXiv2006

Generalized Doubling Constructions for Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then one can show that a perturbation of this approximate hypersurface exists which satisfies the CMC condition. The results of this paper generalize those of the authors in math.DG/0511742.

math.DG↗

arXiv2006

Doubling Constant Mean Curvature Tori in the 3-Sphere

The Clifford tori in the 3-sphere are a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) surfaces. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each sub-lattice point; and then one can show that a perturbation of this approximate submanifold exists which satisfies the CMC condition.

math.DG↗

arXiv2006

Perturbative Solutions of the Extended Constraint Equations in General Relativity

The extended constraint equations arise as a special case of the conformal constraint equations that are satisfied by an initial data hypersurface $Z$ in an asymptotically simple spacetime satisfying the vacuum conformal Einstein equations developed by H. Friedrich. The extended constraint equations consist of a quasi-linear system of partial differential equations for the induced metric, the second fundamental form and two other tensorial quantities defined on $Z$, and are equivalent to the usual constraint equations that $Z$ satisfies as a spacelike hypersurface in a spacetime satisfying Einstein's vacuum equation. This article develops a method for finding perturbative, asymptotically flat solutions of the extended constraint equations in a neighbourhood of the flat solution on Euclidean space. This method is fundamentally different from the `classical' method of Lichnerowicz and York that is used to solve the usual constraint equations.

gr-qc↗

arXiv2002

Exploring the Conformal Constraint Equations

One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where the conformal factor vanishes, namely at the boundary representing null infinity. This problem can be avoided by means of a technique of H. Friedrich, which replaces the Einstein equations in the unphysical spacetime by an equivalent system of equations which is regular at the boundary. The initial value problem for these equations produces a system of constraint equations known as the conformal constraint equations. This work describes some of the properties of the conformal constraint equations and develops a perturbative method of generating solutions near flat space under certain simplifying assumptions.

gr-qc↗

arXiv2002

Deformations of minimal Lagrangian submanifolds with boundary

Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold $M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is shown how deformations of $L$ which keep the boundary of $L$ confined to $W$ can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near $L$ with boundary on $W$ is found to be finite dimensional and is parametrised over the space of harmonic 1-forms of $L$ satisfying Neumann boundary conditions. The second is that if $W'$ is a symplectic, codimension 2 submanifold sufficiently near $W$, then under suitable conditions, there exists a minimal Lagrangian submanifold $L'$ near $L$ with boundary on $W'$.

math.DG↗

arXiv2002

Regularizing a singular special Lagrangian variety

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension $n=3$). Then, $M_{1} \cup M_{2}$ is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds $M_α$ with boundary that converges to $M_{1} \cup M_{2}$ in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of $M_{1} \cap M_{2}$ and then by perturbing this approximate solution until it becomes minimal and Lagrangian.

math.DG↗