2020 | *Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary* | Lashi Bandara, Medet Nursultanov, Julie RowlettZeitschrift: To appear in Annali della Scuola Normale Superiore di Pisa. Classe di ScienzeLink zum Preprint

### Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

#### Autoren: Lashi Bandara, Medet Nursultanov, Julie Rowlett
(2020)

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

Zeitschrift:

To appear in Annali della Scuola Normale Superiore di Pisa. Classe di Scienze

2020 | *Heat kernels and regularity for rough metrics on smooth manifolds* | Lashi Bandara, Paul BryanZeitschrift: Mathematische NachrichtenReihe: 293Seiten: 2255-2270Link zur Publikation
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Link zum Preprint

### Heat kernels and regularity for rough metrics on smooth manifolds

#### Autoren: Lashi Bandara, Paul Bryan
(2020)

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

Zeitschrift:

Mathematische Nachrichten

2019 | *Functional calculus and harmonic analysis in geometry* | Lashi BandaraZeitschrift: São Paulo Journal of Mathematical Sciences, Special Section: An Homage to Manfredo P. do CarmoSeiten: 1-34Link zur Publikation
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### Functional calculus and harmonic analysis in geometry

#### Autoren: Lashi Bandara
(2019)

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.

Zeitschrift:

São Paulo Journal of Mathematical Sciences, Special Section: An Homage to Manfredo P. do Carmo

2019 | *Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions* | Lashi Bandara, Andreas RosénZeitschrift: Comm. Part. Diff. Eq.Verlag: Taylor & FrancisBand: DOI: 10.1080/03605302.2019.1611847Link zur Publikation
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### Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions

#### Autoren: Lashi Bandara, Andreas Rosén
(2019)

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator D_{B} in L^{2} depends Riesz continuously on L^{∞} perturbations of local boundary conditions B. The Lipschitz bound for the map B→D_{B}(1+D_{B}^{2})^{-1/2} depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Zeitschrift:

Comm. Part. Diff. Eq.

Band:

DOI: 10.1080/03605302.2019.1611847

2019 | *Boundary value problems for general first-order elliptic differential operators* | Christian Bär, Lashi BandaraLink zum Preprint

### Boundary value problems for general first-order elliptic differential operators

#### Autoren: Christian Bär, Lashi Bandara
(2019)

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.

We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We provide examples which are conveniently treated by our methods.

2018 | *ShapeRotator: An R tool for standardized rigid rotations of articulated three‐dimensional structures with application for geometric morphometrics* | Marta Vidal-García, Lashi Bandara, J. Scott KeoghZeitschrift: Ecology and EvolutionVerlag: John Wiley & Sons Ltd.Seiten: 4669-4675Band: 2018;8Link zur Publikation

### ShapeRotator: An R tool for standardized rigid rotations of articulated three‐dimensional structures with application for geometric morphometrics

#### Autoren: Marta Vidal-García, Lashi Bandara, J. Scott Keogh
(2018)

The quantification of complex morphological patterns typically involves comprehensive shape and size analyses, usually obtained by gathering morphological data from all the structures that capture the phenotypic diversity of an organism or object. Articulated structures are a critical component of overall phenotypic diversity, but data gathered from these structures are difficult to incorporate into modern analyses because of the complexities associated with jointly quantifying 3D shape in multiple structures. While there are existing methods for analyzing shape variation in articulated structures in two‐dimensional (2D) space, these methods do not work in 3D, a rapidly growing area of capability and research. Here, we describe a simple geometric rigid rotation approach that removes the effect of random translation and rotation, enabling the morphological analysis of 3D articulated structures. Our method is based on Cartesian coordinates in 3D space, so it can be applied to any morphometric problem that also uses 3D coordinates (e.g., spherical harmonics). We demonstrate the method by applying it to a landmark‐based dataset for analyzing shape variation using geometric morphometrics. We have developed an R tool (ShapeRotator) so that the method can be easily implemented in the commonly used R package *geomorph* and *MorphoJ* software. This method will be a valuable tool for 3D morphological analyses in articulated structures by allowing an exhaustive examination of shape and size diversity.

Zeitschrift:

Ecology and Evolution

Verlag:

John Wiley & Sons Ltd.

2017 | *Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics* | Lashi Bandara, Hemanth SaratchandranZeitschrift: Journal of Functional AnalysisReihe: 273Seiten: 3719-3758Link zur Publikation
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### Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

#### Autoren: Lashi Bandara, Hemanth Saratchandran
(2017)

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth.

Zeitschrift:

Journal of Functional Analysis

2017 | *Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric* | Lashi Bandara, Alan McIntosh, Andreas RosénZeitschrift: Math. Ann.Verlag: SpringerSeiten: 1-53Link zur Publikation
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Link zum Preprint

### Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric

#### Autoren: Lashi Bandara, Alan McIntosh, Andreas Rosén
(2017)

We prove that the Atiyah-Singer Dirac operator D_{g} in L^{2} depends Riesz continuously on L^{∞} perturbations of complete metrics g on a smooth manifold. The Lipschitz bound for the map g→D_{g}(1+D_{g}^{2})^{-1/2} depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

2017 | *Geometric singularities and a flow tangent to the Ricci flow* | Lashi Bandara, Sajjad Lakzian, Michael MunnZeitschrift: Ann. Sc. Norm. Super. Pisa Cl. Sci.Seiten: 763-804Band: 17, no. 2Link zur Publikation
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### Geometric singularities and a flow tangent to the Ricci flow

#### Autoren: Lashi Bandara, Sajjad Lakzian, Michael Munn
(2017)

We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangential to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non-singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel.

When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.

Zeitschrift:

Ann. Sc. Norm. Super. Pisa Cl. Sci.

2017 | *Continuity of solutions to space-varying pointwise linear elliptic equations* | Lashi BandaraZeitschrift: Publ. Mat.Seiten: 239-258Band: 61, no. 1Link zur Publikation
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Link zum Preprint

### Continuity of solutions to space-varying pointwise linear elliptic equations

#### Autoren: Lashi Bandara
(2017)

We consider pointwise linear elliptic equations of the form L_{x}u_{x}=η_{x} on a smooth compact manifold where the operators L_{x} are in divergence form with real, bounded, measurable coefficients that vary in the space variable x. We establish L^{2}-continuity of the solution at x whenever the coefficients of L_{x} are L^{∞}-continuous at x and the initial datum is L^{2}-continuous at x. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics g<ub>t that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on M with a C^{1} heat kernel on a "non-singular" nonempty open subset N, we show that x x↦g_{t}(x) is continuous whenever x ∈ N.

2016 | *The Kato square root problem on vector bundles with generalised bounded geometry* | Lashi Bandara, Alan McIntoshZeitschrift: J. Geom. Anal.Verlag: SpringerSeiten: 428-462Band: 26, no. 1Link zur Publikation
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### The Kato square root problem on vector bundles with generalised bounded geometry

#### Autoren: Lashi Bandara, Alan McIntosh
(2016)

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive operators over the bundle of finite rank tensors. These results are obtained as a special case of similar estimates on smooth vector bundles satisfying a criterion which we call generalised bounded geometry. We prove this by establishing quadratic estimates for perturbations of Dirac type operators on such bundles under an appropriate set of assumptions.

Zeitschrift:

J. Geom. Anal.

2016 | *Rough metrics on manifolds and quadratic estimates* | Lashi Bandara,Zeitschrift: Math. Z.Verlag: SpringerSeiten: 1245-1281Band: 283, no. 3-4Link zur Publikation
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Link zum Preprint

### Rough metrics on manifolds and quadratic estimates

#### Autoren: Lashi Bandara,
(2016)

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a continuous metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.

2015 | *Self-adjointness of the Gaffney Laplacian on vector bundles* | Lashi Bandara, Ognjen MilatovicZeitschrift: Math. Phys. Anal. Geom.Verlag: SpringerSeiten: 14 pp.Band: 18, no. 1, Art. 17Link zur Publikation
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### Self-adjointness of the Gaffney Laplacian on vector bundles

#### Autoren: Lashi Bandara, Ognjen Milatovic
(2015)

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

Zeitschrift:

Math. Phys. Anal. Geom.

2014 | *Density problems on vector bundles and manifolds* | Lashi BandaraZeitschrift: Proc. Amer. Math. Soc.Verlag: AMSSeiten: 2683-2695Band: 142, no. 8Link zur Publikation
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### Density problems on vector bundles and manifolds

#### Autoren: Lashi Bandara
(2014)

We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.

Zeitschrift:

Proc. Amer. Math. Soc.

2013 | *Square roots of perturbed subelliptic operators on Lie groups* | Lashi Bandara, A. F. M. ter Elst, Alan McIntoshZeitschrift: Studia Math.Seiten: 193-217Band: 216, no. 3Link zur Publikation
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### Square roots of perturbed subelliptic operators on Lie groups

#### Autoren: Lashi Bandara, A. F. M. ter Elst, Alan McIntosh
(2013)

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, then we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small perturbations of the coefficients.

2012 | *Real Harmonic Analysis, Lectures by Pascal Auscher with the assistance of Lashi Bandara* | Pascal Auscher, Lashi BandaraVerlag: Australian National University PressLink zur Publikation

### Real Harmonic Analysis, Lectures by Pascal Auscher with the assistance of Lashi Bandara

#### Autoren: Pascal Auscher, Lashi Bandara
(2012)

This book presents the material covered in graduate lectures delivered at The Australian National University in 2010. Real Harmonic Analysis originates from the seminal works of Zygmund and Calderón, pursued by Stein, Weiss, Fefferman, Coifman, Meyer and many others. Moving from the classical periodic setting to the real line, then to higher dimensional Euclidean spaces and finally to, nowadays, sets with minimal structures, the theory has reached a high level of applicability. This is why it is called real harmonic analysis: the usual exponential functions have disappeared from the picture. Set and function decomposition prevail.

Verlag:

Australian National University Press

2011 | *Quadratic Estimates and perturbations of Dirac type operators on doubling measure metric spaces* | Lashi BandaraReihe: Proc. Centre Math. Appl. Austral. Nat. Univ.Verlag: Austral. Nat. Univ., CanberraBuchtitel: Proceedings of the AMSI International Conference on Harmonic Analysis and ApplicationsSeiten: 1-21Band: 45Link zur Publikation
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### Quadratic Estimates and perturbations of Dirac type operators on doubling measure metric spaces

#### Autoren: Lashi Bandara
(2011)

We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these operators have a bounded functional calculus. In particular, we deduce a Kato square root type estimate.

Reihe:

Proc. Centre Math. Appl. Austral. Nat. Univ.

Verlag:

Austral. Nat. Univ., Canberra

Buchtitel:

Proceedings of the AMSI International Conference on Harmonic Analysis and Applications