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David Wiygul

Publications and source records attributed to David Wiygul.

15 recordsLinked to original sources
arXiv2026

Morse index of Karcher saddle towers in $\mathbb{R}^2 \times \mathbb{S}^1(m)$

For each integer $k \geq 3$ Hermann Karcher identified a complete singly periodic minimal surface $\Xi_k$ (unique up to similarity) with $2k$ ends asymptotic to the union of $k$ planes intersecting equiangularly along a single line and with genus zero in the quotient by a fundamental translation. Writing $\Xi_{k,m}$ for the quotient of $\Xi_k$ by translation through $m \geq 1$ fundamental periods, we study the Morse index and nullity of the subfamilies $\Xi_{k,2}$ and $\Xi_{3,m}$. In the $m=2$ case we prove for all $k \geq 3$ that $\Xi_{k,2}$ has Morse index $4k-3$ and nullity $3$. In the $k=3$ case we prove that there exists a real number $\alpha^* \in (1/3,1/2)$ such that for all $m \geq 1$ the Morse index of $\Xi_{3,m}$ is $6m- 4 \lfloor m\alpha^* \rfloor - 3$ and its nullity is $3$ unless $m\alpha^*$ is an integer, in which case its nullity is $7$. Both proofs exploit the symmetries of each surface and the Dirichlet-to-Neumann map on a half period to reduce the problem to Fourier-analytic computations on the unit circle (after a conformal change).

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arXiv2025

A new family of minimal surfaces of even genus in the three-dimensional sphere

We discover a family of closed, embedded minimal surfaces in the three-dimensional round sphere which includes new examples with low genus. The existence proof relies on an equivariant min-max procedure applied to a novel sweepout which is constructed by fusing the equatorial sphere with the Clifford torus. We determine the full symmetry groups of our surfaces, prove lower bounds on their Morse indices, and show that they are geometrically distinct from all previously known examples.

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arXiv2025

Disc stackings and their Morse index

We construct free boundary minimal disc stackings, with any number of strata, in the three-dimensional Euclidean unit ball, and prove uniform, linear lower and upper bounds on the Morse index of all such surfaces. Among other things, our work implies for any positive integer $k$ the existence of $k$-tuples of distinct, pairwise non-congruent, embedded free boundary minimal surfaces all having the same topological type. In addition, since we prove that the equivariant Morse index of any such free boundary minimal stacking, with respect to its maximal symmetry group, is bounded from below by (the integer part of) half the number of layers, it follows that any possible realization of such surfaces via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters. This also shows that it is only for $N=2$ and $N=3$ layers that free boundary minimal disc stackings can be obtained by means of one-dimensional mountain pass schemes.

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arXiv2024

Index growth not imputable to topology

We employ partitioning methods, in the spirit of Montiel--Ros but here recast for general actions of compact Lie groups, to prove effective lower bounds on the Morse index of certain families of closed minimal hypersurfaces in the round four-dimensional sphere, and of free boundary minimal hypersurfaces in the Euclidean four-dimensional ball. Our analysis reveals, in particular, phenomena of linear index growth for sequences of minimal hypersurfaces of fixed topological type, in strong contrast to the three-dimensional scenario.

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arXiv2024

New low-genus desingularizations of three Clifford tori and related characterizations

For each nonnegative integer $m$ we construct in the round three-sphere a closed embedded minimal surface of genus $48m+25$ which can be interpreted as a desingularization of the union of three Clifford tori intersecting pairwise orthogonally, along a total of six great circles. Each such surface is generated, under the action of a group of symmetries, by a disc with hexagonal boundary, all of whose sides are contained in great circles. We prove a uniqueness result for this disc, and, as a corollary, we characterize these surfaces. This characterization implies that similar surfaces we constructed for sufficiently high $m$ by gluing methods, in an earlier article, coincide with the ones here. For low $m$ the surfaces constructed here are new. Similarly, we prove uniqueness of the generating discs for one of two families constructed by Choe and Soret (namely the surfaces they call odd) and show that these surfaces also coincide, when of sufficiently high genus, with surfaces we have constructed by gluing.

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arXiv2024

Second-order mass estimates for static vacuum metrics with small Bartnik data

Given on the $2$-sphere Bartnik data (prescribed metric and mean curvature) that is a small perturbation of the corresponding data for the standard unit sphere in Euclidean space, we estimate to second order, in the size of the perturbation, the mass of the asymptotically flat static vacuum extension (unique up to diffeomorphism) which is a small perturbation of the flat metric on the exterior of the unit ball in Euclidean space and induces the prescribed data on the boundary sphere. As an application we obtain a new upper bound on the Bartnik mass of small metric spheres to fifth order in the radius.

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arXiv2024

Spectral estimates for free boundary minimal surfaces via Montiel-Ros partitioning methods

We adapt and extend the Montiel-Ros methodology to compact manifolds with boundary, allowing for mixed (including oblique) boundary conditions and also accounting for the action of a finite group $G$ together with an additional twisting homomorphism $σ\colon G\to\operatorname O(1)$. We then apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate of the Morse index of free boundary minimal surfaces with respect to the topological data (i. e. the genus and the number of boundary components) of the surfaces in question. In particular, we compute the exact values of the equivariant Morse index and nullity for two infinite families of examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided linear bounds when the equivariance constraint is lifted.

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arXiv2019

The index and nullity of the Lawson surfaces $ξ_{g,1}$

We prove that the Lawson surface $ξ_{g,1}$ in Lawson's original notation, which has genus $g$ and can be viewed as a desingularization of two orthogonal great two-spheres in the round three-sphere ${\mathbb{S}}^3$, has index $2g+3$ and nullity $6$ for any genus $g\ge2$. In particular $ξ_{g,1}$ has no exceptional Jacobi fields, which means that it cannot `flap its wings' at the linearized level and is $C^1$-isolated.

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arXiv2018

Minimal surfaces in the 3-sphere by stacking Clifford tori

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by $k\ell m^2 (N-1)$ small catenoidal tunnels, with $k \ell m^2$ tunnels joining each pair of neighboring tori. In the large-$m$ limit for fixed $N$, $k$, and $\ell$, the corresponding surfaces converge to $\mathbb{T}$ counted with multiplicity $N$.

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arXiv2017

The Bartnik-Bray outer mass of small metric spheres in time-symmetric 3-slices

Given a sphere with Bartnik data close to that of a round sphere in Euclidean 3-space, we compute its Bartnik-Bray outer mass to first order in the data's deviation from the standard sphere. The Hawking mass gives a well-known lower bound, and an upper bound is obtained by estimating the mass of a static vacuum extension. As an application we confirm that in a time-symmetric slice concentric geodesic balls shrinking to a point have mass-to-volume ratio converging to the energy density at their center, in accord with physical expectation and the behavior of other quasilocal masses. For balls shrinking to a flat point we can also compute the outer mass to fifth order in the radius---the term is proportional to the Laplacian of the scalar curvature at the center---but our estimate is not refined enough to identify this term at a point which is merely scalar flat. In particular it cannot discern gravitational contributions to the mass.

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arXiv2017

Minimal surfaces in the 3-sphere by desingularizing intersecting Clifford tori

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an additional Clifford torus equidistant from the original two circles of intersection, so that the latter torus orthogonally intersects each of the former $k$ tori along a pair of disjoint orthogonal circles, near which the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type resemble surfaces constructed by Choe and Soret \cite{CS} by different methods where the number of handles desingularizing each circle is the same. There is a plethora of new examples which are more complicated and on which the number of handles for the two circles differs. Examples of the second type are new as well.

math.DG↗
arXiv2015

Minimal surfaces in the 3-sphere by doubling the Clifford torus over rectangular lattices

Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point of the torus and genus tending to infinity. Each surface in a given sequence resembles a pair of tori close to the limit torus and joined by many catenoidal bridges arranged over a rectangular lattice on the limit. The collection of sequences is indexed by the ratio of the lengths of the lattice edges, which may be any prescribed positive rational. Unlike the surfaces of Kapouleas and Yang, these new embeddings are not symmetric with respect to any isometries of the 3-sphere exchanging the two sides of the limit Clifford torus, except when the corresponding lattice is square.

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