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You searched for +publisher:"Brown University" +contributor:("Daskalopoulos, Georgios"). Showing records 1 – 2 of 2 total matches.

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1. Freidin, Brian. Harmonic maps between and into singular spaces.

Degree: Department of Mathematics, 2018, Brown University

We study the roles of domain and target curvatures in harmonic maps into metric spaces with upper curvature bounds. We begin computing the domain and target variation formulas, incorporating the roles of the domain Ricci and scalar curvatures in the former, and the target's curvature bound in the latter. These formulas are optimal in the sense that they match the classical formulas in the case when the target is a smooth manifold. We then use these formulas to derive a generalization of the Eells-Sampson Bochner formula for this case of singular targets, generalizing the work of Chen who proved a subharmonicity statement under stronger geometric assumptions. Our formula includes terms involving domain Ricci and target curvature terms that are analogous to terms from the Eells-Sampson formula. As a consequence of this formula we derive rigidity results for harmonic maps, including reproving a result of Korevaar and Schoen on rigidity of harmonic maps from flat tori to non-positively curved spaces. We derive more consequences from our domain and target variation formulas. First, we conclude that the distance squared of the map to a point is a weakly subharmonic function if the image is convex, and use this, along with a reverse Poincar\'e inequality, to prove a Liouville-type theorem for harmonic maps from Riemannian polyhedra to spaces with positive curvature bounds. We also analyze conformal harmonic maps to singular spaces; for domains of dimension at least 3, we generalize a result from the theory of harmonic morphisms, that such maps are rescaled isometric immersions. In dimension 2 we generalize a well known result that gives an a-priori energy bound for conformal harmonic maps from hyperbolic surfaces to spaces with curvature bounded above by a negative constant. Advisors/Committee Members: Daskalopoulos, Georgios (Advisor), Kapouleas, Nicolaos (Reader), Mese, Chikako (Reader).

Subjects/Keywords: Geometric analysis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Freidin, B. (2018). Harmonic maps between and into singular spaces. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:792815/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Freidin, Brian. “Harmonic maps between and into singular spaces.” 2018. Thesis, Brown University. Accessed July 15, 2019. https://repository.library.brown.edu/studio/item/bdr:792815/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Freidin, Brian. “Harmonic maps between and into singular spaces.” 2018. Web. 15 Jul 2019.

Vancouver:

Freidin B. Harmonic maps between and into singular spaces. [Internet] [Thesis]. Brown University; 2018. [cited 2019 Jul 15]. Available from: https://repository.library.brown.edu/studio/item/bdr:792815/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Freidin B. Harmonic maps between and into singular spaces. [Thesis]. Brown University; 2018. Available from: https://repository.library.brown.edu/studio/item/bdr:792815/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Chen, Qile. Logarithmic stable maps to Deligne-Faltings pairs.

Degree: PhD, Mathematics, 2011, Brown University

The main topic of this dissertation is to introduce the notion of minimal log stable maps, which gives a new compactification of the space of stable maps relative to a Cartier divisor. A stable map relative to a Cartier divisor D is a holomorphic map from a nodal algebraic curve to a projective variety with prescribed tangency multiplicities along the marked points with respect to D. Rather than using the expanded degenerations, we adopt the tool of logarithmic geometry in the sense of Kato-Fontaine-Illusie. We first define the notion of log stable maps over schemes by equipping both source curves and the target of the usual stable maps with log structures, and show that the stack of log stable maps over schemes is algebraic. The log structures on the stable maps allow us to keep track of the tangency conditions even if the underlying map is degenerated. However, the stack of log stable maps over schemes is too large, and fails to be of finite type. Minimality is introduced to select a smaller open substack of the stack of log stable maps. The stack of minimal log stable maps is shown to be proper and Deligne-Mumford. Furthermore, the stack with its natural minimal log structure represents the category of log stable maps over fine and saturated log schemes. The representability allows us to generalize our construction to the case of generalized Deligne-Faltings pairs. In particular, this covers the case of stable maps relative to a simple normal crossings divisor.This is in part a joint work with my advisor Dan Abramovich. Advisors/Committee Members: Abramovich, Dan (Director), Daskalopoulos, Georgios (Reader), Gillam, William (Reader), Maulik, Davesh (Reader).

Subjects/Keywords: log stable map; minimal log stable map

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Chen, Q. (2011). Logarithmic stable maps to Deligne-Faltings pairs. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11240/

Chicago Manual of Style (16th Edition):

Chen, Qile. “Logarithmic stable maps to Deligne-Faltings pairs.” 2011. Doctoral Dissertation, Brown University. Accessed July 15, 2019. https://repository.library.brown.edu/studio/item/bdr:11240/.

MLA Handbook (7th Edition):

Chen, Qile. “Logarithmic stable maps to Deligne-Faltings pairs.” 2011. Web. 15 Jul 2019.

Vancouver:

Chen Q. Logarithmic stable maps to Deligne-Faltings pairs. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2019 Jul 15]. Available from: https://repository.library.brown.edu/studio/item/bdr:11240/.

Council of Science Editors:

Chen Q. Logarithmic stable maps to Deligne-Faltings pairs. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11240/

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