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1. CJM Online first

Glöckner, Helge
Completeness of infinite-dimensional Lie groups in their left uniformity
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.

Keywords:infinite-dimensional Lie group, left uniform structure, completeness
Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05

2. CJM Online first

Georgescu, Magdalena Cecilia
Integral Formula for Spectral Flow for $p$-Summable Operators
Fix a von Neumann algebra $\mathcal{N}$ equipped with a suitable trace $\tau$. For a path of self-adjoint Breuer-Fredholm operators, the spectral flow measures the net amount of spectrum which moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer-Fredholm operator affiliated with $\mathcal{N}$. If the unbounded operator is p-summable (that is, its resolvents are contained in the ideal $L^p$), then it is possible to obtain an integral formula which calculates spectral flow. This integral formula was first proven by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\theta$-summable operators, and then using Laplace transforms to obtain a p-summable formula. In this paper, we present a direct proof of the p-summable formula, which is both shorter and simpler than theirs.

Keywords:spectral flow, $p$-summable Fredholm module
Categories:19k56, 46L87, , 58B34

3. CJM 2016 (vol 68 pp. 841)

Gupta, Sanjiv Kumar; Hare, Kathryn
Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra
Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$ and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank $+1$ in the case of type $A_{n}$). More recently, the minimal integer $k=k(X)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\in \mathfrak{g}$. In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $(X_{1},\dots,X_{L})$, with $X_{i}\in \mathfrak{g},$ which have the property that the convolution of the $L$-orbital measures supported on the orbits generated by the $X_{i}$ is absolutely continuous and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $\mathfrak{g}$ and the structure of the annihilating roots of the $X_{i}$. Such a characterization was previously known only for type $A_{n}$.

Keywords:compact Lie algebra, orbital measure, absolutely continuous measure
Categories:43A80, 17B45, 58C35

4. CJM 2016 (vol 69 pp. 241)

Adamus, Janusz; Seyedinejad, Hadi
Finite Determinacy and Stability of Flatness of Analytic Mappings
It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

Keywords:finite determinacy, stability, flatness, openness, complete intersection
Categories:58K40, 58K25, 32S05, 58K20, 32S30, 32B99, 32C05, 13B40

5. CJM 2014 (vol 67 pp. 107)

Chang, Jui-En; Xiao, Ling
The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space
In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

Categories:53A99, 5J15, 58J05

6. CJM 2012 (vol 65 pp. 1255)

Iglesias-Zemmour, Patrick
Variations of Integrals in Diffeology
We establish the formula for the variation of integrals of differential forms on cubic chains, in the context of diffeological spaces. Then, we establish the diffeological version of Stoke's theorem, and we apply that to get the diffeological variant of the Cartan-Lie formula. Still in the context of Cartan-De-Rham calculus in diffeology, we construct a Chain-Homotopy Operator $\mathbf K$ we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the Chain-Homotopy Operator which used in symplectic diffeology to construct the Moment Map.

Keywords:diffeology, differential geometry, Cartan-De-Rham calculus
Categories:58A10, 58A12, 58A40

7. CJM 2012 (vol 65 pp. 879)

Kawabe, Hiroko
A Space of Harmonic Maps from the Sphere into the Complex Projective Space
Guest-Ohnita and Crawford have shown the path-connectedness of the space of harmonic maps from $S^2$ to $\mathbf{C} P^n$ of a fixed degree and energy.It is well-known that the $\partial$ transform is defined on this space. In this paper,we will show that the space is decomposed into mutually disjoint connected subspaces on which $\partial$ is homeomorphic.

Keywords:harmonic maps, harmonic sequences, gluing
Categories:58E20, 58D15

8. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants.

Keywords:higher order forms, iterated integrals
Categories:14F35, 11F12, 55D35, 58A10

9. CJM 2011 (vol 64 pp. 924)

McCann, Robert J.; Pass, Brendan; Warren, Micah
Rectifiability of Optimal Transportation Plans
The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Categories:49K20, 49K60, 35J96, 58C07

10. CJM 2011 (vol 63 pp. 721)

Autin, Aymeric
Isoresonant Complex-valued Potentials and Symmetries
Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(\Delta-z)^{-1}$, $z\in\mathbb{C} \setminus \mathbb{R}^+$, has a meromorphic continuation through $\mathbb{R}^+$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $\Delta+V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian.

Categories:31C12, 58J50

11. CJM 2010 (vol 63 pp. 55)

Chau, Albert; Tam, Luen-Fai; Yu, Chengjie
Pseudolocality for the Ricci Flow and Applications
Perelman established a differential Li--Yau--Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flat manifolds. We also prove a long time existence result for the K\"ahler--Ricci flow on complete nonnegatively curved K\"ahler manifolds.

Categories:53C44, 58J37, 35B35

12. CJM 2010 (vol 62 pp. 1264)

Chen, Jingyi; Fraser, Ailana
Holomorphic variations of minimal disks with boundary on a Lagrangian surface
Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n+\mu(E,F)$ if $u$ is unstable.

Categories:58E12, 53C21, 53C26

13. CJM 2010 (vol 62 pp. 1325)

Mo, Xiaohuan; Yu, Changtao
On Some Explicit Constructions of Finsler Metrics with Scalar Flag Curvature
We give an explicit construction of polynomial (\emph{of arbitrary degree}) $(\alpha,\beta)$-metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all non-trivial projectively flat $(\alpha,\beta)$-metrics of constant flag curvature.

Keywords:Finsler metric, scalar curvature, projective flatness

14. CJM 2009 (vol 62 pp. 242)

Azagra, Daniel; Fry, Robb
A Second Order Smooth Variational Principle on Riemannian Manifolds
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

Keywords:smooth variational principle, Riemannian manifold
Categories:58E30, 49J52, 46T05, 47J30, 58B20

15. CJM 2009 (vol 62 pp. 52)

Deng, Shaoqiang
An Algebraic Approach to Weakly Symmetric Finsler Spaces
In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions $2$ and $3$. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing S-curvature may exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, S-curvature
Categories:53C60, 58B20, 22E46, 22E60

16. CJM 2009 (vol 61 pp. 548)

Girouard, Alexandre
Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces
We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

Categories:35P, 58J

17. CJM 2008 (vol 60 pp. 1336)

Olver, Peter J.; Pohjanpelto, Juha
Moving Frames for Lie Pseudo--Groups
We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

Categories:58A15, 58A20, 58H05, 58J70

18. CJM 2008 (vol 60 pp. 1149)

Petersen, Kathleen L.; Sinclair, Christopher D.
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Categories:11C08, 28A75, 15A52, 54H10, 58D19

19. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms
Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition
Categories:31C25, 35B50, 60J45, 35J, 53C, 58

20. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point
This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.

Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity
Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

21. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

22. CJM 2008 (vol 60 pp. 297)

Bini, G.; Goulden, I. P.; Jackson, D. M.
Transitive Factorizations in the Hyperoctahedral Group
The classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type~$A$ to other finite reflection groups and, in particular, to type~$B$. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an $\gS_2$-symmetry. The type~$A$ case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type~$B$ case that is studied here.

Categories:05A15, 14H10, 58D29

23. CJM 2008 (vol 60 pp. 241)

Alexandrova, Ivana
Semi-Classical Wavefront Set and Fourier Integral Operators
Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions.

Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysis
Categories:35S30, 35A27, 58J40, 81Q20

24. CJM 2007 (vol 59 pp. 1245)

Chen, Qun; Zhou, Zhen-Rong
On Gap Properties and Instabilities of $p$-Yang--Mills Fields
We consider the $p$-Yang--Mills functional $(p\geq 2)$ defined as $\YM_p(\nabla):=\frac 1 p \int_M \|\rn\|^p$. We call critical points of $\YM_p(\cdot)$ the $p$-Yang--Mills connections, and the associated curvature $\rn$ the $p$-Yang--Mills fields. In this paper, we prove gap properties and instability theorems for $p$-Yang--Mills fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.

Keywords:$p$-Yang--Mills field, gap property, instability, submanifold
Categories:58E15, 53C05

25. CJM 2007 (vol 59 pp. 943)

Finster, Felix; Kraus, Margarita
A Weighted $L^2$-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds
We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold~$(M^n, g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of~$M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

Categories:83C60, 35Q75, 35J45, 58J05
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