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The Geometry of Differential Harnack Estimates

Sebastian Helmensdorfer, Peter Topping (2011/2012)

Séminaire de théorie spectrale et géométrie

In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind the Harnack...

The geometry of null systems, Jordan algebras and von Staudt's theorem

Wolfgang Bertram (2003)

Annales de l’institut Fourier

We characterize an important class of generalized projective geometries ( X , X ' ) by the following essentially equivalent properties: (1) ( X , X ' ) admits a central null-system; (2) ( X , X ' ) admits inner polarities: (3) ( X , X ' ) is associated to a unital Jordan algebra. These geometries, called of the first kind, play in the category of generalized projective geometries a rôle comparable to the one of the projective line in the category of ordinary projective geometries. In this general set-up, we prove an analogue of von Staudt’s...

The graded differential geometry of mixed symmetry tensors

Andrew James Bruce, Eduardo Ibarguengoytia (2019)

Archivum Mathematicum

We show how the theory of ℤ 2 n -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

The gradient flow of Higgs pairs

Jiayu Li, Xi Zhang (2011)

Journal of the European Mathematical Society

We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle ( E , H 0 ) over a Kähler surface ( M , ω ) , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition ( A 0 , φ 0 ) converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point ( A ∞ , φ ∞ ) of this functional. We also prove that the limiting Higgs pair ( A ∞ , φ ∞ ) can be extended smoothly to a vector bundle E ∞ over...

The graph of a totally geodesic foliation

Robert A. Wolak (1995)

Annales Polonici Mathematici

We study the properties of the graph of a totally geodesic foliation. We limit our considerations to basic properties of the graph, and from them we derive several interesting corollaries on the structure of leaves.

The Group of Invertible Elements of the Algebra of Quaternions

Irina A. Kuzmina, Marie Chodorová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We have, that all two-dimensional subspaces of the algebra of quaternions, containing a unit, are 2-dimensional subalgebras isomorphic to the algebra ℂ of complex numbers. It was proved in the papers of N. E. Belova. In the present article we consider a 2-dimensional subalgebra ( i ) of complex numbers with basis 1 , i and we construct the principal locally trivial bundle which is isomorphic to the Hopf fibration.

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