arXiv subjects

Manousos Maridakis

Publications and source records attributed to Manousos Maridakis.

5 recordsLinked to original sources

arXiv2022

A Localization Theorem for Dirac operators

We study perturbed Dirac operators of the form $ D_s= D + s\A :Γ(E^0)\rightarrow Γ(E^1)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps $\A : E^0\rightarrow E^1$ for $s>>0$. Under a simple algebraic criterion on the pair $(c, \A)$, solutions of $D_sψ=0$ concentrate as $s\to\infty$ around the singular set $Z_\A$ of $\A$. We prove a spectral separation property of the deformed Laplacians $D_s^*D_s$ and $D_s D_s^*$, for $s>>0$. As a corollary we prove an index localization theorem.

math.DG↗

arXiv2019

Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article arXiv:1903.01953, we apply our abstract Lojasiewicz-Simon gradient inequalities to prove a Lojasiewicz-Simon gradient inequalities for the harmonic map energy functional using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities for the harmonic map energy functional generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997). In our monograph arXiv:1510.03815, we prove Lojasiewicz--Simon gradient inequalities for coupled Yang--Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang--Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of arbitrary dimension and due to Rade (1992) for dimensions two and three.

math.DG↗

arXiv2019

Lojasiewicz--Simon gradient inequalities for the harmonic map energy function

We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient inequalities for the harmonic map energy function generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997).

math.DG↗

arXiv2019

Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions

In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions generalize that of the pure Yang-Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of dimensions two, three and four and due to Rade (1992) for dimensions two and three.

math.DG↗

arXiv2015

Spinor Pairs and the Concentration Principle for Dirac operators

We study perturbed Dirac operators of the form $ D_s= D + s{\cal A} :Γ(E)\rightarrow Γ(F)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps ${\cal A} : E\rightarrow F$ for $s>>0$. Under a simple algebraic criterion on the pair $(c, {\cal A})$, solutions of $D_sψ=0$ concentrate as $s\to\infty$ around the singular set $Z_\A\subset X$ of ${\cal A}$. We give many examples, the most interesting ones arising from a general ``spinor pair'' construction.

math.DG↗