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Nikolaos Kapouleas

Publications and source records attributed to Nikolaos Kapouleas.

18 recordsLinked to original sources
arXiv2025

Index and nullity of minimal surface doublings, I

We prove that for any large enough $m \in\mathbb{N}$, the genus $γ=m+1$ equator-poles minimal surface doubling of the equatorial two-sphere $Σ^0 = \mathbb{S}^2_{\mathrm{eq}}$ in the round three-sphere $\mathbb{S}^3$, which has two catenoidal bridges at the poles and $m$ bridges equidistributed along the equatorial circle $\mathscr{C}$ of $Σ^0 $ and was discovered in earlier work of Kapouleas, has index $2γ+5=2m+7$ and nullity $6$, and so it has no exceptional Jacobi fields and is $C^1$-isolated.

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arXiv2024

New minimal surface doublings of the Clifford torus and contributions to questions of Yau

The purpose of this article is three-fold. First, we apply a general theorem from our earlier work to produce many new minimal doublings of the Clifford Torus in the round three-sphere. This construction generalizes and unifies prior doubling constructions for the Clifford Torus, producing doublings with catenoidal bridges arranged along parallel copies of torus knots. Ketover has also constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but his methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while ours are not constrained this way. Second, we use this family to prove a new, quadratic lower bound for the number of embedded minimal surfaces in $\mathbb{S}^3$ with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in $\mathbb{S}^3$ with fixed genus. Third, we verify Yau's conjecture for the first eigenvalue of minimal surfaces in $\mathbb{S}^3$ in the following cases. First, for all minimal surface doublings of the equatorial two-sphere constructible by our earlier general theorem. Second, for all the Clifford Torus doublings constructed in this article.

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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

Minimal hypersurfaces in $\mathbb{S}^{4}(1)$ by doubling the equatorial $\mathbb{S}^{3}$

For each large enough $m\in\mathbb{N}$ we construct by PDE gluing methods a closed embedded smooth minimal hypersurface ${\breve{M}_m}$ doubling the equatorial three-sphere $\mathbb{S}_{\mathrm{eq}}^3$ in $\mathbb{S}^4(1)$, with ${\breve{M}_m}$ containing $m^2$ bridges modelled after the three-dimensional catenoid and centered at the points of a square $m\times m$ lattice $L$ contained in the Clifford torus $\mathbb{T}^2\subset \mathbb{S}_{\mathrm{eq}}^3$. This answers a long-standing question of Yau in the case of $\mathbb{S}^4(1)$ and long-standing questions of Hsiang. Similarly we construct a self-shrinker ${\breve{M}_{\mathrm{shr},m}}$ of the Mean Curvature Flow in $\mathbb{R}^4$ doubling the three-dimensional spherical self-shrinker $\mathbb{S}_{\mathrm{shr}}^3\subset \mathbb{R}^4$ with the bridges centered at the points of a square $m\times m$ lattice $L$ contained in a Clifford torus $\mathbb{T}^2\subset \mathbb{S}_{\mathrm{shr}}^3$. Both constructions respect the symmetries of the lattice $L$ as a subset of $\mathbb{S}^4(1)$ or $\mathbb{R}^4$ and are based on the Linearized Doubling (LD) methodology which was first introduced in the construction of minimal surface doublings of $\mathbb{S}_{\mathrm{eq}}^2$ in $\mathbb{S}^3(1)$. Furthermore $\breve{M}_m$ converges as $m \to\infty$ in the varifold sense to $2\mathbb{S}_{\mathrm{eq}}^3$, and its volume $|\breve{M}_m| < 2|\mathbb{S}_{\mathrm{eq}}^3|$.

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arXiv2022

Free Boundary Minimal Annuli Immersed in the Unit 3-Ball

Using the linearized doubling methodology we carry out a PDE gluing construction of a discrete family of non-rotational properly immersed free boundary minimal annuli in the Euclidean unit 3-ball. The surfaces we construct resemble equatorial disks joined by half-catenoidal bridges at the boundary.

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arXiv2022

Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers

In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Σ$, a minimal surface $\breve{M}$ resembling two copies of $Σ$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $φ$ on $Σ$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|\breve{M}|$ of $\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|\breve{M}| < 2 | Σ|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)\times \mathbb{Z}_2)$-symmetric backgrounds $(Σ, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Σ=\mathbb{S}^2 \subset N=\mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.

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arXiv2021

Free Boundary Minimal surfaces in the Euclidean Three-Ball close to the boundary

We construct free boundary minimal surfaces (FBMS) embedded in the unit ball in the Euclidean three-space which are compact, lie arbitrarily close to the boundary unit sphere, are of genus zero, and their boundary has an arbitrarily large number of connected boundary components. The construction is by PDE gluing methods and the surfaces are desingularizations of unions of many catenoidal annuli and two flat discs. The union of the boundaries of the catenoidal annuli and discs is the union of a large finite number of parallel circles contained in the unit sphere, with each parallel circle contained in the boundary of exactly two of the catenoidal annuli and discs.

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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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arXiv2017

Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa-Hoffman-Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover using the min-max method.

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arXiv2017

Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the Linearized Doubling (LD) methodology developed before and the LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides (work in progress). In particular they allow us to develop in this forthcoming work doubling constructions for the catenoid in Euclidean three-space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow. Our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be "densely distributed", that is do not miss any open set of $\mathbb{S}^2_{eq}$ in the limit. This in particular leads to interesting observations which seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point.

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arXiv2017

Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC $n$-hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter $\underlineτ$. This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given $k\ge2$ and $n\ge3$ it allows us to realize infinitely many topological types as CMC $n$-hypersurfaces in $\mathbb R^{n+1}$ with $k$ ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when $k\ge3$ where the number of possible topological types for each $k$ is finite but tends to $\infty$ as $k\to\infty$.

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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.

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arXiv2016

Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere, I

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions we refine and reorganize the doubling methodology in ways which we expect to apply also to further constructions. In particular we introduce what we call linearized doubling, which is an intermediate step where singular solutions to the linearized equation are constructed subject to appropriate linear and nonlinear conditions. Linearized doubling provides a systematic approach for dealing with the obstructions involved and also understanding in detail the regions further away from the catenoidal bridges.

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arXiv2014

Mean curvature self-shrinkers of high genus: Non-compact examples

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in $\mathbb{R}^3$ over a $2π/(g+1)$-periodic graph on an equator of the unit sphere $\mathbb{S}^2\subseteq\mathbb{R}^3$, with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

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arXiv2012

Embedded Constant Mean Curvature Surfaces in Euclidean Three Space

In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (1995). As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in Kapouleas (1990) and we produce a very large class of new embedded examples of finite topology.

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arXiv2010

Doubling and Desingularization Constructions for Minimal Surfaces

In the first part of the paper we discuss the current status of the application of the gluing methodology to doubling and desingularization constructions for minimal surfaces in Riemannian three-manifolds. In particular a doubling construction for equatorial spheres in $S^3(1)$ is announced. Aspects of the current understanding of existence and uniqueness questions for closed minimal embedded surfaces in $S^3(1)$ are also discussed, and some new uniqueness questions are proposed. In the second part of the paper we discuss some of the ideas and provide an outline for a general desingularization construction without imposed symmetries. This paper is the author's contribution to the volume in honor of Professor Richard M. Schoen's sixtieth birthday.

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arXiv2010

Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones

We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, in preparation for their application to our higher dimensional special Legendrian gluing constructions. In particular the symmetries of these cones and their asymptotics near the spherical limit are analysed. All $\sorth{p} \times \sorth{q}$-invariant SL cones arise from a more general construction of independent interest which we call the special Legendrian twisted product construction. Using this twisted product construction and simple variants of it we can construct a constellation of new special Lagrangian and Hamiltonian stationary cones in $\C^{n}$. We prove the following theorems: A. there are infinitely many topological types of special Lagrangian and Hamiltonian stationary cones in $\C^{n}$ for all $n\ge 4$, B. for $n\ge 4$ special Lagrangian and Hamiltonian stationary torus cones in $\C^{n}$ can occur in continuous families of arbitrarily high dimension and C. for $n\ge 6$ there are infinitely many topological types of special Lagrangian and Hamiltonian stationary cones in $\C^{n}$ that can occur in continuous families of arbitrarily high dimension.

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