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Paul M. N. Feehan

Publications and source records attributed to Paul M. N. Feehan.

At least 19 recordsLinked to original sources
arXiv2026

Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

We previously developed an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept of virtual Morse-Bott indices for singular critical points of Hamiltonian functions for the induced circle actions (see Feehan, arXiv:2206.14710). For Hamiltonian functions of circle actions on closed, complex Kaehler manifolds, the virtual Morse-Bott index coincides with the classical Morse-Bott index due to Bott (1954) and Frankel (1959). A key principle in our approach is that positivity of the virtual Morse-Bott index at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when that critical point is a singular point in the moduli space. In this monograph, we consider our method in the context of the moduli space of non-Abelian monopoles over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.

math.DG↗
arXiv2025

Almost Hermitian structures on moduli spaces of non-Abelian monopoles and applications to the topology of symplectic four-manifolds

This work is a sequel to our previous monograph arXiv:2010.15789 (to appear in AMS Memoirs), where we initiated our program to prove that the Bogomolov-Miyaoka-Yau inequality holds for closed, symplectic four-manifolds and, more generally, for closed, smooth four-manifolds with a Seiberg-Witten basic class. This inequality was first proved for compact, complex surfaces of general type independently by Miyaoka and Yau in 1977. Our approach uses a version of Morse theory for a natural Hamiltonian, the square of the $L^2$ norm of the coupled spinors, for the circle action on the moduli space of non-Abelian monopoles over a closed four-manifold. It has the aim of proving the existence of a projectively anti-self-dual connection on a rank-two Hermitian vector bundle over a blow-up of the four-manifold, where the first Pontrjagin number of the vector bundle is negative and greater than or equal to minus the Euler characteristic of the blown-up four-manifold. Our Morse theory argument relies on positivity of virtual Morse-Bott indices for critical points of Hamiltonians for circle actions on complex analytic spaces (or real analytic spaces that, locally, are sufficiently well-approximated by complex analytic model spaces), as developed by the first author in arXiv:2206.14710. In our application to the moduli space of non-Abelian monopoles, the critical points are fixed points of the circle action and thus represented by Seiberg-Witten monopoles.

math.DG↗
arXiv2024

Morse theory for the Yang-Mills energy function near flat connections

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $d\geq 2$ is bounded by a constant times the $L^p$ norm of the curvature, $\|F_A\|_{L^p(X)}$, when $G$ is a compact Lie group, $F_A$ is $L^p$-small, and $p>d/2$. While we prove that this estimate holds when the Yang-Mills energy function on the space of Sobolev connections is Morse-Bott along the moduli subspace $M(P)$ of flat connections, it does not hold when the Yang-Mills energy function fails to be Morse-Bott, such as at the product connection in the moduli space of flat $\mathrm{SU}(2)$ connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck's estimate always holds provided one replaces $\|F_A\|_{L^p(X)}$ by a suitable power $\|F_A\|_{L^p(X)}^λ$, where the positive exponent $λ$ reflects the structure of non-regular points in $M(P)$. The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of Sobolev connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. A special case of our estimate, when $X$ has dimension four and the connection $A$ is anti-self-dual, was proved by Fukaya (1998) by entirely different methods. Lastly, we prove that if $A$ is a smooth Yang-Mills connection with small enough energy, then $A$ is necessarily flat.

math.DG↗
arXiv2022

Bialynicki-Birula theory, Morse-Bott theory, and resolution of singularities for analytic spaces

Our goal in this work is to develop aspects of Bialynicki-Birula and Morse-Bott theory that can be extended from the classical setting of smooth manifolds to that of complex analytic spaces with a holomorphic $\mathbb{C}^*$ action. We extend prior results on existence of Bialynicki-Birula decompositions for compact, complex Kähler manifolds to non-compact complex manifolds and develop functorial properties of the Bialynicki-Birula decomposition, in particular with respect to blowup along a $\mathbb{C}^*$-invariant, embedded complex submanifold. We deduce the existence of a Bialynicki-Birula decomposition for a $\mathbb{C}^*$-invariant, closed, complex analytic subspace of complex manifold with a $\mathbb{C}^*$ action; derive geometric consequences for the positivity of the Bialynicki-Birula nullity, co-index, and index at a fixed point; and we develop stronger versions of these results by applying resolution of singularities for analytic spaces.

math.CV↗
arXiv2020

Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Lojasiewicz-Simon gradient inequality for abstract Morse-Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis in arXiv:1510.03817. We apply this result to prove the optimal Lojasiewicz-Simon gradient inequality for the self-dual Yang-Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang-Mills energy function over closed Riemannian manifolds of dimension $d \geq 2$, when known to be Morse-Bott at a given Yang-Mills connection. We also prove the optimal Lojasiewicz-Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map. We also prove the Morse-Bott property for irreducible Yang-Mills $U(n)$ connections over Riemann surfaces and hence a new proof of the optimal Lojasiewicz-Simon gradient inequality for such critical points.

math.DG↗
arXiv2020

Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, $Au := -\mathrm{tr}(aD^2u)- + cu$, with partial Dirichlet boundary conditions. The coefficient, $a(x)$, is assumed to vanish along a non-empty open subset, $\partial_0\mathscr{O}$, called the \emph{degenerate boundary portion}, of the boundary, $\partial\mathscr{O}$, of the domain $\mathscr{O}\subset\mathbb{R}^d$, while $a(x)$ is non-zero at any point of the \emph{non-degenerate boundary portion}, $\partial_1\mathscr{O} := \partial\mathscr{O}\setminus\overline{\partial_0\mathscr{O}}$. If an $A$-subharmonic function, $u$ in $C^2(\mathscr{O})$ or $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$, is $C^1$ up to $\partial_0\mathscr{O}$ and has a strict local maximum at a point in $\partial_0\mathscr{O}$, we show that $u$ can be perturbed, by the addition of a suitable function $w\in C^2(\mathscr{O})\cap C^1(\mathbb{R}^d)$, to a strictly $A$-subharmonic function $v=u+w$ having a local maximum in the interior of $\mathscr{O}$. Consequently, we obtain strong and weak maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$ and $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ which are $C^1$ up to $\partial_0\mathscr{O}$. Only the non-degenerate boundary portion, $\partial_1\mathscr{O}$, is required for boundary comparisons. Our results extend those in Daskalopoulos and Hamilton (1998), Epstein and Mazzeo [arXiv:1110.0032], and the author [arXiv:1204.6613, 1306.5197], where $\mathrm{tr}(aD^2u)$ is in addition assumed to be continuous up to and vanish along $\partial_0\mathscr{O}$ in order to yield comparable maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$, while the results developed here for $A$-subharmonic functions in $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ are entirely new.

math.AP↗
arXiv2020

On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space and the Morse Lemma for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Lojasiewicz-Simon gradient inequalities for analytic functions on Banach spaces.

math.DG↗
arXiv2020

Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of the Lojasiewicz inequalities in the special case where the function is $C^1$ with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real or complex analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Lojasiewicz inequalities when a function is $C^N$ and generalized Morse-Bott of order $N \geq 3$; we gave an elementary proof of the Lojasiewicz inequalities when a function is $C^2$ and Morse-Bott in arXiv:1708.09775v4 (finite-dimensional case) and arXiv:1706.09349 (infinite-dimensional case).

math.DG↗
arXiv2019

Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory

We describe a new approach to the problem of constructing gluing parameterizations for open neighborhoods of boundary points of moduli spaces of anti-self-dual connections over closed four-dimensional manifolds. Our approach employs general results from differential topology for $C^1$ maps of smooth Banach manifolds with corners, providing a method that should apply to other problems in geometric analysis involving the gluing construction of solutions to nonlinear partial differential equations.

math.DG↗
arXiv2019

The SO(3) monopole cobordism and superconformal simple type

We show that the SO(3) monopole cobordism formula from Feehan and Leness (2002) implies that all smooth, closed, oriented four-manifolds with $b^1=0$ and $b^+\geq 3$ and odd with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999) This implies the lower bound, conjectured by Fintushel and Stern (2001) on the number of Seiberg-Witten basic classes in terms of topological data.

math.DG↗
arXiv2019

Superconformal simple type and Witten's conjecture

Let $X$ be a smooth, closed, connected, orientable four-manifold with $b^1(X)=0$ and $b^+(X)\geq 3$ and odd. We show that if $X$ has Seiberg-Witten simple type, then the SO(3)-monopole cobordism formula of Feehan and Leness (2002) implies Witten's Conjecture relating the Donaldson and Seiberg-Witten invariants.

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

Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to remove a positivity constraint on a combination of the Ricci and Riemannian curvatures in a previous $L^{d/2}$-energy gap result due to Gerhardt (2010) and a previous $L^\infty$-energy gap result due to Bourguignon, Lawson, and Simons (1981, 1979), as well as an $L^2$-energy gap result due to Nakajima (1987) for a Yang-Mills connection over the sphere, $S^d$, but with an arbitrary Riemannian metric. The main correction in this version involves replacement of the role of Corollary 4.3 due to Uhlenbeck (1985) and Theorem 5.1 due to the author in the published version of this article at http://dx.doi.org/10.1016/j.aim.2017.03.023 by that of Theorems 1 and 9 due to the author in arXiv:1906.03954.

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↗
arXiv2018

An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032]. In this monograph, we prove --- modulo a gluing theorem which is an extension of our earlier work in [arXiv:math/9907107] --- that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. This conclusion is analogous to the Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson invariants of a four-manifold with $b_2^+=1$; that wall-crossing formula and the resulting structure of Donaldson invariants for four-manifolds with $b_2^+=1$ were established, assuming the Kotschick-Morgan conjecture, by Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier [arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the Kotschick-Morgan conjecture to an extension of previously established gluing theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and references therein). Since the first version of our monograph was circulated, applications of our results have appeared in the proof of Property P for knots by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson invariants for symplectic four-manifolds [arXiv:1301.0377].

math.DG↗
arXiv2016

Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riemannian manifolds of arbitrary dimension and apply the resulting framework to prove new results for the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, smooth Riemannian manifolds, including the following. If the initial connection is close enough to a local minimum of the Yang-Mills energy functional, in a norm sense when the base manifold has arbitrary dimension or in an energy sense when the base manifold has dimension four, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. If the initial connection is allowed to have arbitrary energy but we restrict to the setting of a Hermitian vector bundle over a compact, complex, Hermitian (but not necessarily Kaehler) surface and the initial connection has curvature of type (1,1), then the Yang-Mills gradient flow exists for all time, though bubble singularities may (and in certain cases must) occur in the limit as time tends to infinity.

math.DG↗