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

Degree: Department of Mathematics, 2018, Brown University

URL: https://repository.library.brown.edu/studio/item/bdr:792815/

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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/

Not specified: Masters Thesis or Doctoral Dissertation

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

Degree: PhD, Mathematics, 2011, Brown University

URL: https://repository.library.brown.edu/studio/item/bdr:11240/

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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/