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Mario B. Schulz

Publications and source records attributed to Mario B. Schulz.

16 recordsLinked to original sources
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.

math.DG↗
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.

math.DG↗
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.

math.DG↗
arXiv2024

Free boundary minimal Möbius bands in toroids

We prove that strictly mean convex toroids contain infinitely many (geometrically distinct) embedded free boundary minimal Möbius bands as well as infinitely many embedded free boundary minimal annuli. The surfaces in both families are constructed by means of equivariant variational methods and their areas grow linearly with the order of their symmetry groups.

math.DG↗
arXiv2024

Genus one critical catenoid

We use variational methods to construct a free boundary minimal surface in the three-dimensional unit ball with genus one, two boundary components and prismatic symmetry. Key ingredients are an extension of the equivariant min-max theory to include orientation-reversing isometries and the discovery of a nontrivial two-parameter sweepout.

math.DG↗
arXiv2024

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

math.DG↗
arXiv2024

Equivariant free boundary minimal discs and annuli in ellipsoids

We employ equivariant variational methods to construct new examples of nonplanar free boundary minimal discs in ellipsoids. We also prove that every ellipsoid contains at least three distinct embedded free boundary minimal annuli with dihedral symmetry.

math.DG↗
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.

math.DG↗
arXiv2023

Minimal hypertori in the four-dimensional sphere

We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as infinitely many, pairwise non-isometric, immersed ones. Our analysis also yields infinitely many, pairwise non-isometric, minimally embedded hyperspheres and thus provides a self-contained solution to Chern's spherical Bernstein conjecture in dimensions four and six.

math.DG↗
arXiv2023

Topological control for min-max free boundary minimal surfaces

We establish general bounds on the topology of free boundary minimal surfaces obtained via min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We prove that the first Betti number is lower semicontinuous along min-max sequences converging in the sense of varifolds to free boundary minimal surfaces. In the orientable case, we obtain an even stronger result which implies that if the number of boundary components increases in the varifold limit, then the genus decreases at least as much. We also present several compelling applications, such as the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.

math.DG↗
arXiv2019

Instantaneously complete Yamabe flow on hyperbolic space

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.

math.AP↗
arXiv2019

Incomplete Yamabe flows and removable singularities

We study the Yamabe flow on a Riemannian manifold of dimension $m\geq3$ minus a closed submanifold of dimension $n$ and prove that there exists an instantaneously complete solution if and only if $n>\frac{m-2}{2}$. In the remaining cases $0\leq n\leq\frac{m-2}{2}$ including the borderline case, we show that the removability of the $n$-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.

math.DG↗
arXiv2019

Unconditional existence of conformally hyperbolic Yamabe flows

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.

math.AP↗
arXiv2018

Yamabe flow on non-compact manifolds with unbounded initial curvature

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.

math.AP↗