scholarly journals Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

2019 ◽  
pp. 1-29 ◽  
Author(s):  
Mikhail G. Katz ◽  
Stéphane Sabourau
Keyword(s):  
Total Curvature ◽  
Nonpositive Curvature ◽  
Difficult Problem ◽  
Conical Singularities ◽  
Natural Approach ◽  
Nonpositively Curved ◽  
Hands On ◽  
Genus Formula ◽  
Generalized Metrics ◽  
Asymptotic Upper Bound

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by Gromov in 1983, who proposed an argument toward the existence of [Formula: see text]-extremizers exploiting the theory of [Formula: see text]-regularity developed by White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don’t have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. This result exploits a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, as well as the combinatorial/topological estimates of Malestein–Rivin–Theran, Przytycki, Aougab–Biringer–Gaster and Greene on the number of curves meeting at most once, combined with a kite excision move. The move merges pairs of conical singularities on a surface of genus [Formula: see text] and leads to an asymptotic upper bound [Formula: see text] on the number of singularities.

TRIANGLES, SQUARES AND GEODESICS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250041 ◽  
Author(s):  
RENA LEVITT ◽  
JON MCCAMMOND
Keyword(s):  
Nonpositive Curvature ◽  
Higher Dimensions ◽  
Fundamental Groups ◽  
Nonpositively Curved ◽  
Cube Complexes

In the early 1990s Steve Gersten and Hamish Short proved that compact nonpositively curved triangle complexes have biautomatic fundamental groups and that compact nonpositively curved square complexes have biautomatic fundamental groups. In this paper we report on the extent to which results such as these extend to nonpositively curved complexes built out a mixture of triangles and squares. Since both results by Gersten and Short have been generalized to higher dimensions, this can be viewed as a first step towards unifying Januszkiewicz and Świȧtkowski's theory of simplicial nonpositive curvature with the theory of nonpositively curved cube complexes.

scholarly journals Boundaries of nonpositively curved groups of the form G × ℤn

1996 ◽  
Vol 38 (2) ◽  
pp. 177-189 ◽  
Author(s):  
Philip L. Bowers ◽  
Kim Ruane
Keyword(s):  
Nonpositive Curvature ◽  
Hyperbolic Groups ◽  
Combinatorial Group Theory ◽  
Discrete Groups ◽  
The Past ◽  
Negatively Curved ◽  
Automatic Groups ◽  
Nonpositively Curved ◽  
Positively Curved ◽  
Combinatorial Group

The introduction of curvature considerations in the past decade into Combinatorial Group Theory has had a profound effect on the study of infinite discrete groups. In particular, the theory of negatively curved groups has enjoyed significant and extensive development since Cannon's seminal study of cocompact hyperbolic groups in the early eighties [7]. Unarguably the greatest influence on the direction of this development has been Gromov's tour de force, his foundational essay in [12] entitled Hyperbolic Groups. Therein Gromov hints at the prospect of developing a corresponding theory of “non-positively curved groups” in his non-definition (Gromov's terminology) of a semihyperbolic group as a group that “looks as if it admits a discrete co-compact isometric action on a space of nonpositive curvature”; [12, p. 81]. Such a development is now occurring and is closely related to the other notable outgrowth of the theory of negatively curved groups, that of automatic groups [10]; we mention here the works [3] and [6] as developments of a theory of nonpositively curved groups along with Chapter 6 of Gromov's more recent treatise [13]. A natural question that serves both to guide and organize the developing theory is: to what extent is the well-developed theory of negatively curved groups reflected in and subsumed under the developing theory of nonpositively curved groups? Our overall interest is in one aspect of this question—namely, as the question relates to the boundaries of groups and spaces: can one define the boundary of a nonpositively curved group intrinsically in a way that generalizes that of negatively curved groups and retains some of their essential features?

scholarly journals Uniformization of branched surfaces and Higgs bundles

2021 ◽  
pp. 2150096
Author(s):  
Indranil Biswas ◽  
Steven Bradlow ◽  
Sorin Dumitrescu ◽  
Sebastian Heller
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Cone Angle ◽  
Higgs Bundles ◽  
Constant Negative Curvature ◽  
Conical Singularities ◽  
Genus Formula ◽  
Conical Metric ◽  
Branched Surfaces ◽  
Cone Metric

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

scholarly journals Controlled Homeomorphisms Over Nonpositively Curved Manifolds

2000 ◽  
Vol 43 (3) ◽  
pp. 343-354
Author(s):  
Bruce Hughes ◽  
Larry Taylor ◽  
Bruce Williams
Keyword(s):  
Nonpositive Curvature ◽  
Curved Manifolds ◽  
Nonpositively Curved

AbstractWe obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

A systolic-like extremal genus two surface

2019 ◽  
Vol 11 (03) ◽  
pp. 721-738
Author(s):  
Stéphane Sabourau ◽  
Zeina Yassine
Keyword(s):  
Homotopy Classes ◽  
Conical Singularities ◽  
Nonpositively Curved ◽  
Flat Metric ◽  
Genus Two ◽  
Systolic Inequality

It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes.

The Instructional SEM Laboratory at Iowa State University

1996 ◽  
Vol 54 ◽  
pp. 396-397
Author(s):  
L. S. Chumbley ◽  
M. Meyer ◽  
K. Fredrickson ◽  
F.C. Laabs
Keyword(s):  
Materials Science ◽  
Computer Network ◽  
Energy Dispersive Spectrometer ◽  
Computer Control ◽  
State University ◽  
Iowa State University ◽  
Internet Server ◽  
Schematic Layout ◽  
Hands On ◽  
Improve Instruction

The Materials Science Department at Iowa State University has developed a laboratory designed to improve instruction in the use of the scanning electron microscope (SEM). The laboratory makes use of a computer network and a series of remote workstations in a classroom setting to provide students with increased hands-on access to the SEM. The laboratory has also been equipped such that distance learning via the internet can be achieved.A view of the laboratory is shown in Figure 1. The laboratory consists of a JEOL 6100 SEM, a Macintosh Quadra computer that acts as a server for the network and controls the energy dispersive spectrometer (EDS), four Macintosh computers that act as remote workstations, and a fifth Macintosh that acts as an internet server. A schematic layout of the classroom is shown in Figure 2. The workstations are connected directly to the SEM to allow joystick and computer control of the microscope. An ethernet connection between the Quadra and the workstations allows students seated there to operate the EDS. Control of the microscope and joystick is passed between the workstations by a switch-box assembly that resides at the microscope console. When the switch-box assembly is activated a direct serial line is established between the specified workstation and the microscope via the SEM’s RS-232.

Students Involvement in Multilingual and Multicultural Community Services

2015 ◽  
Vol 22 (2) ◽  
pp. 77-89 ◽  
Author(s):  
Ying-Chiao Tsao
Keyword(s):  
Service Learning ◽  
Cultural Competence ◽  
Linguistically Diverse ◽  
Community Services ◽  
Cultural Humility ◽  
Self Assessment ◽  
Multicultural Community ◽  
Raising Awareness ◽  
Hands On ◽  
Teaching Learning

Promoting cultural competence in serving diverse clients has become critically important across disciplines. Yet, progress has been limited in raising awareness and sensitivity. Tervalon and Murray-Garcia (1998) believed that cultural competence can only be truly achieved through critical self-assessment, recognition of limits, and ongoing acquisition of knowledge (known as “cultural humility”). Teaching cultural humility, and the value associated with it remains a challenging task for many educators. Challenges inherent in such instruction stem from lack of resources/known strategies as well as learner and instructor readiness. Kirk (2007) further indicates that providing feedback on one's integrity could be threatening. In current study, both traditional classroom-based teaching pedagogy and hands-on community engagement were reviewed. To bridge a gap between academic teaching/learning and real world situations, the author proposed service learning as a means to teach cultural humility and empower students with confidence in serving clients from culturally/linguistically diverse backgrounds. To provide a class of 51 students with multicultural and multilingual community service experience, the author partnered with the Tzu-Chi Foundation (an international nonprofit organization). In this article, the results, strengths, and limitations of this service learning project are discussed.

A ’Hands-On’ Approach

ASHA Leader ◽  
2012 ◽  
Vol 17 (9) ◽  
pp. 55-55
Author(s):  
Kimberly Abts
Keyword(s):  
Hands On

Telepractice: A Survey of AuD Students Pre- and Post-Telepractice

2017 ◽  
Vol 2 (18) ◽  
pp. 28-41
Author(s):  
Kelli M. Watts ◽  
Laura B. Willis
Keyword(s):  
Doctoral Students ◽  
Graduate School ◽  
Professional Services ◽  
Hands On ◽  
Before And After ◽  
Lack Of Knowledge

Telepractice, defined by the American Speech-Language-Hearing Association (ASHA, n.d.) as “the application of telecommunications technology to the delivery of professional services at a distance by linking clinician to client, or clinician to clinician, for assessment, intervention, and/or consultation,” is a quickly growing aspect of practicing audiology. However, only 12% of audiologists are involved in providing services via telepractice (REDA International, Inc., 2002). Lack of knowledge regarding telepractice has been cited as one of the reasons many audiologists do not use telepractice to provide audiology services. This study surveyed audiology doctoral students regarding their opinions about the use of telepractice both before and after their opportunity to provide services via telepractice sessions. The authors expected that by providing students the opportunity to have hands-on training in telepractice with supervision, they would be more open to using telepractice after becoming licensed audiologists. Overall, the data indicates benefits of exposing students to telepractice while they are in graduate school.

Sign in / Sign up

Export Citation Format

Share Document