Smooth Minimal Surface Research Articles

An extreme limit with nonnegative scalar curvature

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. Here we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the W1,p sense for p<2 to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. This paper only requires expertise in smooth Riemannian Geometry, smooth minimal surfaces, and Sobolev Spaces. In a second paper, requiring expertise in metric geometry, the first two authors prove intrinsic flat and Gromov–Hausdorff convergence of our sequence to this extreme limit space and investigate its geometric properties.

Extra Connected Components in the Moduli Space of Stable Surfaces

Abstract We construct examples of non-smoothable stable surface with invariants in a wide range comprising all possible invariants of smooth minimal surfaces of general type but non-standard second plurigenus. We deduce that the moduli space of stable surfaces $\overline {{\mathfrak {M}}}_{k,l}$ has at least $k-l+2$ connected components distinguished by the second plurigenus and, in particular, always has more irreducible components than the Gieseker moduli space with the same invariants.

Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities

By employing the method of moving planes in a novel way we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.

Application Assessments of Using Scarp Boundary-Fitted, Volume Constrained, Smooth Minimal Surfaces as Failure Interfaces of Deep-Seated Landslides

More than 9000 potential deep-seated landslide sites in the mountain ranges of Taiwan have been identified by a series of renewed governmental hazard mitigation initiatives after the 2009 Morakot typhoon. Among these sites, 186 sites have protection targets where thorough mitigation strategies are to be implemented. One of the important tasks in the hazard mitigation initiative is to estimate the volume, fracture interface and related quantities of each landslide site. In addition, with this number of sites, an automated tool is needed to generate predictions at low operational costs. We propose to use volume-constrained smooth minimal surfaces to approximate the landslide fracture interfaces. A volume-constrained smooth minimal surface in the current context is defined as a differentiable surface that encloses a given landslide volume with the minimal surface area. Although the stratigraphy and geological structures are omitted, the smooth minimal surface method is verified with 24 known landslides and is shown to be able to generate acceptable, approximated fracture interfaces. A collection of assessment indices is employed to measure the fitness of the predictions. Finally, the prediction fitness versus the landslide scarp geometry is investigated.

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

AbstractIn a previous paper, new closed embedded smooth minimal surfaces in the round three‐sphere were constructed, each resembling two parallel copies of the equatorial two‐sphere 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 in the previous paper. The LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides [15]. In particular, they allow us to develop 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.Unlike in the previous paper, our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed,” that is, they do not miss any open set of in the limit. This in particular leads to interesting observations that 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. © 2019 Wiley Periodicals, Inc.

The maintenance of inserted titanium implants: in-vitro evaluation of exposed surfaces cleaned with three different instruments.

Changes to titanium implants smooth-surfaces after instrumentation were comparatively analyzed using low-vacuum scanning electron microscopy (LV-SEM) and white-light confocal (WLC) profilometry, to accurately evaluate curved surfaces. Sixty titanium implants screwed to their abutments were randomly split into three groups for cleaning treatment with (S) stainless-steel Gracey-curettes, (T) titanium Langer-curettes, and (P) an ultrasonic-device with the probe covered with a plastic-tip. One sector of each implant was left unprocessed (U). The other sectors were cleaned for either 60s, to simulate a single cleaning session, or 180s to simulate a series of sessions. Surface morphology was analyzed by LV-SEM, without metal sputtering. Quantitative evaluations of the roughness of surfaces were performed using a WLC-profilometer. The Wilcoxon and the Mann-Whitney tests were used in statistical comparisons. U-surfaces showed that thin transverse ridges and grooves, i.e. a polarized surface roughness was substantially compromised after S-instrumentation. Small surface alterations, increasing with time, were also recorded after T-·and·P-instrumentation, although to a lesser degree. The gap of the fixture-abutment connection appeared almost completely clean after T-, clotted with titanium debris after S-, and clotted with plastic debris after P-treatment. The mean roughness (Ra) was unchanged after P-, significantly increased after S- and decreased after T-treatment, when compared with U. The Rz roughness-parameter, calculated along the fixture Y-axis, of S, T, and P resulted similar and significantly lower than that of U. Rz (X-axis) resulted unchanged after P-, slightly increased (+40%) after T-, and greatly increased (+260%) after S-treatment, this latter being statistically significant when compared with U. The careful use of titanium-curettes could produce only minimal smooth surface alteration particularly over prolonged treatments, and avoid debris production that could endanger implant preservation.

Cuspidal quintics and surfaces with , and 5-torsion

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

Hyperelliptic surfaces with $K^{2} < 4\chi - 6$

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2 4$ and $K_S^2-3\chi(\mathcal O_S)<-6,$ we compute the maximum value for $\chi(\mathcal O_S).$ This list of possibilities is sharp.

Commuting involutions on surfaces of general type with p g = 0 and K 2 = 7

The aim of this article is to classify the pairs (S, G), where S is a smooth minimal surface of general type with pg = 0 and K2 = 7, G is a subgroup of the automorphism group of S and G is isomorphic to the group \({{\mathbb{Z}}_2^2}\). We show that there are only three possible cases for such pairs. Two of them correspond to known examples, but the existence of the third one remains an open problem. The Inoue surfaces with K2 = 7, which are finite Galois \({\mathbb{Z}_2^2}\)-covers of the 4-nodal cubic surface, are the first examples of such pairs. More recently, the author constructed a new family of such pairs. They are finite Galois \({\mathbb{Z}_2^2}\)-covers of certain 6-nodal Del Pezzo surfaces of degree one. We prove that the base of the Kuranishi family of deformations of a surface in this family is smooth. We show that, in the Gieseker moduli space of canonical models of surfaces of general type, the subset corresponding to the surfaces in this family is an irreducible connected component, normal, unirational of dimension 3.

Automorphisms and quotients of quaternionic fake quadrics

A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some quaternion algebras over real number fields. We provide examples of quaternionic fake quadrics X with a non-trivial automorphism group and compute the invariants of the minimal desingularisation of the quotient of X by this group. In that way we obtain minimal surfaces of general type Z with q=p_g=0 and K^2=4,2 which contain the maximal number of disjoint nodal curves. We then prove that if a surface of general type has the same invariant as Z and same number of nodal curves, we can construct geometrically a surface of general type with K^2=2c_2=8.

Straight ruled surfaces in the Heisenberg group

We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a straight ruled surface, i.e., it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.

A new family of surfaces of general type with $$K^2=7$$ K 2 = 7 and $$p_g=0$$ p g = 0

We construct a new family of smooth minimal surfaces of general type with \(K^2=7\) and \(p_g=0\). We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.

A note on Todorov surfaces

Let S be a Todorov surface, i.e., a minimal smooth surface of general type with q = 0 and pg = 1 having an involution i such that S/i is birational to a K3 surface and such that the bicanonical map of S is composed with i. The main result of this paper is that, if P is the minimal smooth model of S/i, then P is the minimal desingularization of a double cover of P 2 ramified over two cubics. Furthermore it is also shown that, given a Todorov surface S, it is possible to construct Todorov surfaces Sj with K 2 = 1, . . . , K 2 S 1 and such that P is also the smooth minimal model of Sj/ij, where ij is the involution of Sj. Some examples are also given, namely an example different from the examples presented by Todorov in [To2]. 2000 Mathematics Classification: 14J29, 14J28.

C∞-Convergence of Circle Patterns to Minimal Surfaces

AbstractGiven a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

Geography and the Number of Moduli of Surfaces of General Type

The paper considers a relationship between the Chern numbers $K^2_X,\,c_2(X)$ of a smooth minimal surface $X$ of general type and the dimension of the space of infinitesimal deformations of $X$, i.e. $h^1(\Theta_X)$, where $\Theta_X$ is the holomorphic tangent bundle of $X$. We prove that if the ratio of the Chern numbers $\alpha(X) = \frac{c_2(X)}{K^2_X} \leq \frac38$ and $K_X$ is ample then $$h^1(\Theta_X) \leq 9(3c_2-K^2).$$ On the geometric side it is shown that a smooth surface of general type $X$ with $\alpha(X)\leq \frac38$ and $h^1(\Theta_X) \geq 3$ has two distinguished effective divisors $ F$ and $E$ such that $H^1(\Theta_X)$ admits a direct sum decomposition $H^1(\Theta_X) = V_1 \oplus V_0$, where $V_1$ is identified with a subspace of $H^0(\mathcal{O}_X (F))$ while $V_0$ is identified with a subspace of $H^0(\Theta_X \otimes \mathcal{O}_E (E))$. This gives a geometric interpretation of the cohomology classes in $H^1(\Theta_X)$ and allows to bound the dimension of $V_0$ (resp. $V_1$) in terms of geometry of the divisor $E$ (resp. $F$). The main idea of the paper is to use the natural identification $$H^1(\Theta_X) = Ext^1(\Omega_X, \mathcal{O}_X)$$ where $\Omega_X$ is the holomorphic cotangent bundle of $X$. Then the universal extension gives rise to a certain vector bundle whose study constitutes the essential part of the paper.

Discrete constant mean curvature surfaces and their index

We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as discrete minimal catenoids and helicoids. We use these discrete surfaces to study the index of unstable minimal surfaces, by numerically evaluating the spectra of their Jacobi operators. Our numerical estimates confirm known results on the index of some smooth minimal surfaces, and provide additional information regarding their area-reducing variations. The approach here deviates from other numerical investigations in that we add geometric interpretation to the discrete surfaces.

A bound on the plurigenera of projective surfaces

We exhibit a sharp Castelnuovo bound for the ith plurigenus of a smooth minimal surface of general type and of given degree d in the projective space P r , and classify the surfaces attaining the bound, at least when d⪢ r. We give similar results for surfaces not necessarily minimal or of general type, but only for i⪢ r (however, in the case r≤8, we give a complete classification, i.e., for any i≥1). In certain cases (only for r≥12) the surfaces with maximal plurigenus are not Castelnuovo surfaces, i.e., surfaces with maximal geometric genus.