Browsing by Author "Wolf, Michael"
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Item A variational approach to the local uniqueness of minimal surfaces immersed in R(3)(1998) Cunningham, Nancy Elizabeth; Wolf, MichaelIn the past 20 years, many techniques have been developed for proving the existence of complete minimal surfaces immersed in space. Few methods are known for classifying such surfaces. In order to study the structure of spaces of minimal surfaces, we introduce a variational method based in contemporary Teichmuller theory. We apply this method to demonstrate local uniqueness in a model case. We prove that the generalized Chen-Gackstatter surface of genus 2 is locally unique in the space of Weierstrass data of complete, orientable minimal surfaces immersed in space with exact height differential and smallest possible total Gaussian curvature.Item Automorphisms of nonpositively curved cube complexes, right-angled Artin groups and homology(2017-04-20) Bregman, Corey Joseph; Putman, Andrew; Wolf, MichaelRecently, the geometry of CAT(0) cube complexes featured prominently in Agol’s resolution of two longstanding conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. A key step of the proof was to show that every hyperbolic 3-manifold group is virtually special, i.e. virtually the fundamental group of a special nonpositively curved (NPC) cube complex. In this thesis, we study algebraic properties of special groups as they relate to the geometry of special cube complexes. In the first part of the thesis, we introduce a positive integer-valued invariant of special cube complexes called the genus, and show that having genus one is equivalent to having free abelian fundamental group. As a corollary, we obtain a new proof of the fact that every special group is either abelian or surjects onto a non-abelian free group. In the second part of the thesis, we turn our attention to automorphisms of NPC cube complexes. We give a criterion on a special cube complex which implies that any automorphism acts non-trivially on first homology, and show that a non- trivial action on homology can always be achieved by passing to covers. We then apply the criterion to provide a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.Item Costa cousins(1999) Stone, Lorette L.; Wolf, MichaelBryant showed that in a space of constant sectional curvature - c2 (for c ∈ R ), there is a representation for surfaces which have constant mean curvature equal to c, abbreviated as CMC c, which is analogous to the Weierstrass representation in R3 . For c > 0, by rescaling the metric, we can limit ourselves to examining surfaces with CMC 1, immersed in hyperbolic 3-space H3 . Many examples of minimal surfaces in R3 have found their analogs (cousins, in the terminology of Bryant) in the new theory. However there is one minimal surface, a thrice-punctured torus discovered by Costa, whose analog has resisted discovery. In this work we build on the results of Bryant, Umehara and Yamada to show that for a particular thrice punctured torus M, there exists a countable family of complete, finite total curvature, CMC 1, singly branched immersions into H3 having regular ends. We also show that this family is the unique family of such immersions whose Weierstrass data has a certain form. The proof of the existence and uniqueness of this family of singly branched Costa cousins is divided into three parts. In part I we prove that out of a given set of Weierstrass data there is only one real parameter family of candidates which can yield a complete, finite total curvature, possibly singular, CMC 1 immersion. In this case, Bryant's Weierstrass, Representation theorem applies to give us the existence of a multi-valued singular CMC 1 immersion of M into H3. In part II we show that such a singular immersion is well-defined across the handle generators by showing that it can be written as a function of the Weierstrass ℘ -function on M. Finally, in part III, for a countable subset of the real parameter family, we show that such a singular immersion is well-defined in a neighborhood of each of the three ends. From the construction we can conclude that the immersion is a complete, finite total curvature, having regular ends, and having one singularity, which is a branch point.Item Deformations of simply periodic Scherk-type minimal surfaces(2009) McLelland, Matt; Wolf, MichaelWe demonstrate that a certain dimensional family of symmetric singly-periodic minimal surface with Scherk-type ends exists in the neighborhood of a given example if a set of holomorphic quadratic differentials is independent. Our approach extends the work of Traizet, who has previously shown the existence of a family of such minimal surfaces in the neighborhood of a degenerate example consisting of a number of intersecting planes. Whereas in Traizet's construction the underlying conformal structure was a union of Riemann spheres, we treat the case where the underlying conformal structure is a Riemann surface of higher genus. In our approach, admissible surfaces are identified with Weierstrass data satisfying certain constraints. Using the bilinear relations and the Rauch variational formula, we are able to find holomorphic quadratic differentials which represent differentials of the constraints, and whose independence, by an implicit function theorem argument, implies the existence of the desired surface family in a neighborhood of the original. We restrict our attention to tori and develop machinery for investigating the quadratic differentials numerically using interval arithmetic to obtain provable bounds on their residue structure. The developed tools are finally applied to an example surface in Karcher's one-dimensional toroidal saddle tower family, which is shown to exist in a larger three-dimensional family.Item Degeneration of minimal surfaces in the bidisc(2020-04-22) Ouyang, Charles; Wolf, MichaelThis thesis studies the degeneration of a particular class of minimal surfaces in the bidisc, describing both the limiting metric structure and geometry. Minimal surfaces inside symmetric spaces have been shown to be directly related to surface group representations into higher rank Lie groups by recent work of Labourie. Let S be a closed surface of genus g ≥ 2 and let ρ be a maximal PSL(2, R) × PSL(2, R) surface group representation. By a result of Schoen, there is a unique ρ-equivariant minimal surface Σ in H2 × H2. We study the induced metrics on these minimal surfaces and prove the limits are precisely mixed structures. In the second half of the thesis, we provide a geometric interpretation: the minimal surfaces Σ degenerate to the core of a product of two R-trees. As a consequence, we obtain a geometric compactification of the space of maximal representations of π1(S) into PSL(2, R) × PSL(2, R).Item Density of rational points on K3 surfaces over function fields(2012-09-05) Li, Zhiyuan; Hassett, Brendan E.; Wolf, Michael; Rojo, JavierIn this paper, we study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the N´eron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric models are Calabi-Yau. Furthermore, we also generalize our results to the cases of families of higher dimensional Calabi-Yau varieties with Calabi-Yau ambient spaces.Item Flat structures, soap films, and capillary surfaces(2003) Huff, Robert; Wolf, MichaelA technique is presented by way of example for proving the existence of minimal surfaces bounded by straight line segments and planar curves along which the surface meets the plane of the curve at a constant angle. Set in complex analysis, this technique provides a way to construct new examples of soap films and capillary surfaces. The soap films established are a soap film spanning five edges of a regular tetrahedron and a soap film spanning a rectangular prism. The examples of capillary graphs over a square presented here were previously shown to exist by Concus, Finn, and McCuan. However, with this new approach, we are able to examine the behavior of the graphs at the corners of the square. More precisely, we construct two one-parameter families of capillary graphs. The first family provides examples of capillary graphs that have continuous unit normal up to the corner, but the graphing function is not C2 at the corner. The second family consists of capillary graphs with contact angle data in D+2∪D-2 such that the graphing function has a finite jump discontinuity at each corner.Item Geodesic coordinates for the pressure metric at the Fuchsian locus(2020-04-15) Dai, Xian; Wolf, MichaelHigher Teichm{“u}ller theory studies representations of a surface group into a general Lie group that arise as deformation of the classical Teichm{“u}ller space. In this thesis, we focus on the Riemannian geometry for one family of Higher Teichm{“u}ller spaces that are Hitchin components. We study a Riemannian metric, called the pressure metric, in the Hitchin component $\mathcal{H}_{3}(S)$ of surface group representations into $PSL(3,\mathbb{R})$ and prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in $\mathcal{H}_{3}(S)$. The proof is a combination of thermodynamic formalism and Higgs bundle theory. We compute first derivatives of the pressure metric by using Thermodynamic formalism and subshifts of finite type. We then study flat connections from Hitchin’s equations and their parallel transports by invoking a gauge-theoretic formula.Item Handle Addition for Doubly-Periodic Scherk Surfaces(De Gruyter, 2012) Weber, Matthias; Wolf, MichaelWe prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus g with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proof of embeddedness is by the conjugate Plateau method.Item Handle crushing harmonic maps between surfaces(2016-04-20) Huang, Andy C; Wolf, MichaelIn this thesis, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from C (which are conformally once-punctured spheres) to H^2 have been parameterized by holomorphic quadratic differentials on C. Our harmonic maps, mapping a genus g>1 punctured surface to a k-sided polygon, correspond to meromorphic quadratic differentials with one pole of order (k+2) at the puncture and (4g+k−2) zeros (counting multiplicity). In this way, we can associate to these maps a holomorphic quadratic differential on the punctured Riemann surface domain. As an example, we explore a special case of our theorems: the unique harmonic map from a punctured square torus to an ideal square. We use the symmetries of the map to deduce the three possibilities for its Hopf differential.Item Harmonic maps and the geometry of Teichmuller space(2004) Huang, Zheng; Wolf, MichaelIn this thesis work, we investigate the asymptotic behavior of the sectional curvatures of the Weil-Petersson metric on Teichmuller space. It is known that the sectional curvatures are negative. Our method is to investigate harmonic maps from a nearly noded surface to nearby hyperbolic structures, hence to study the Hopf differentials associated to harmonic maps and the analytic formulas resulting from the harmonicity of the maps. Besides providing a quantitative result, our estimates imply that even though the sectional curvatures are negative, they are not staying away from zero. In other words, we show that when the complex dimension of Teichmuller space T is greater than one, then there is no negative upper bound for the sectional curvature of the Weil-Petersson metric. During the proof, we also give the explicit description of a family of tangent planes which are asymptotically flat.Item Harmonic maps of trivalent trees(1991) Stockton, George F.; Wolf, MichaelThis thesis is a study of harmonic maps of trivalent trees into Euclidean space. The existence of such maps is established, and uniqueness is shown to hold up to a certain isotopy condition. Moreover, within its particular isotopy class, each harmonic map is shown to be a local minimum for the energy functional. A harmonic map of a trivalent tree is determined by its associated nodes. Collectively, these nodes are a function of the lengths of the parameter spaces of the paths which comprise the map. It is shown that this node function can be continuously extended to certain parts of the boundary of its domain; these parts of the boundary are closely related to the geometry of the trivalent tree which serves as the domain of the given harmonic map.Item Hitchin Components, Riemannian Metrics and Asymptotics(2014-12-04) Li, Qiongling; Wolf, Michael; Hardt, Robert; Gillman, AdriannaHigher Teichm\"uller spaces are deformation spaces arising from subsets of the space of representations of a surface group into a general Lie group, e.g., $$PSL(n,\RR)$$, which share some of the properties of classical Teichmueller space. By the non-abelian Hodge theory, such representation spaces correspond to the space of Higgs bundles. We focus on two aspects on the Higher Teichm\"uller space: Riemannian geometry and dynamics. First, we construct a new Riemannian metric on the deformation space for $$PSL(3,\RR)$$, and then prove Teichmueller space endowed with Weil-Petersson metric is totally geodesic in deformation space for $$PSL(3,\RR)$$ with the new metric. Secondly, in a joint work with Brian Collier, we are able to obtain asymptotic behaviors and related properties of representations for certain families of Higgs bundles of rank n.Item Holonomy Limits of Cyclic Opers(2016-04-20) Acosta, Jorge A.; Wolf, MichaelGiven a Riemann surface $X = (\Sigma, J)$, we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,t\omega_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tilde{\Sigma},\tilde{J})$ to the symmetric space SL$_n(\mathbb{C}) / \mbox{SU}(n)$ and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of $\omega_n$.Item Infinite Energy Harmonic Maps and Degeneration of Hyperbolic Surfaces in Moduli Space(Journal of Differential Geometry, 1991) Wolf, MichaelItem Limits of minimal surfaces with increasing genus(2007) Kim, Soomin; Wolf, MichaelMinimal Surfaces are surfaces which locally minimize area. These surfaces are well-known as mathematical idealizations of soap films, one area of the calculus of variations which applies to geometric modeling. This thesis is devoted to the clas sification of minimal surfaces, specifically limits of minimal surfaces with increasing genus. In this paper, we will see that a particular well-known family of minimal surfaces, indexed by increasing genus, has a limit, and, further, that limit is nearly a well-known example. This is the first nontrivial example of a limit being taken of a family of minimal surfaces of increasing topological complexity. As a classification result, this would limit the set of possible minimal surfaces, as we would see that new surfaces would not be created through the taking of limits of existing families of surfaces in this way.Item Minimizers of the vector-valued coarea formula(2012-09-05) Carroll, Colin; Hardt, Robert M.; Wolf, Michael; Cox, Steven J.The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.Item Minimizing and flow problems for multiple-valued functions and maps(2007) Zhu, Wei; Hardt, Robert M.; Wolf, MichaelWe consider variational problems in the setting of multiple-valued functions (with a fixed number of values) and multiple-valued maps into manifolds. In particular, for an energy minimizing map into a sphere, we prove that the interior singular set is at least of codimension three. We also construct an energy reducing flow for multiple-valued functions, which is H older continuous with respect to its L 2 norms. Some questions concerning regularity and vanishing of branch points are also addressed.Item On geometry along grafting rays in Teichmuller space(2012-09-05) Laverdiere, Renee; Wolf, Michael; Hardt, Robert M.; Goldman, RonIn this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $$\sigma_{t}$$ at a point along the grafting ray in terms of a conformal factor $$g_{t}$$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.Item Perturbed, Genus One Scherk Surfaces and their limits(2009) Douglas, Casey; Wolf, MichaelThe singly periodic, genus one helicoid is conjectured to be the limit of a one parameter family of doubly periodic minimal surfaces referred to as Perturbed Genus One Scherk Surfaces. Using elementary elliptic function theory, we show such surfaces exist, solving a two-dimensional period problem by perturbing a one-dimensional problem. Using flat structures associated to these minimal surfaces, we then verify the conjecture.