Harmonic Metrics Research Articles

Harmonic metrics for the Hull–Strominger system and stability

We investigate stability conditions related to the existence of solutions of the Hull–Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull–Strominger system is recasted using non-Hermitian Yang–Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull–Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.

On the existence of harmonic metrics on non‐Hermitian Yang–Mills bundles

AbstractIn this paper, we study the non‐Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (Selecta Math. (N.S.) 4 (1998) 279–320).

Integrality of the Betti moduli space

If in a given rank r r , there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number ℓ \ell , there is an absolutely irreducible ℓ \ell -adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r r if once there is an irreducible rank r r complex linear representation with determinant isomorphic to this torsion character, then for any ℓ \ell , there is an absolutely irreducible one of rank r r and determinant this given character, which is defined over Z ¯ ℓ \bar { \mathbb {Z}}_\ell . We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne’s companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong’s conjecture (Gaitsgory for ℓ ≥ 3 \ell \ge 3 ). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple.

Sentinel-2 time series analysis for monitoring multi-taxon biodiversity in mountain beech forests

Biodiversity monitoring represents a major challenge to supporting proper forest ecosystem management and biodiversity conservation. The latter is indeed shifting in recent years from single-species to multi-taxon approaches. However, multi-taxonomic studies are quite rare due to the effort required for performing field surveys. In this context, remote sensing is a powerful tool, continuously providing consistent and open access data at a different range of spatial and temporal scales. In particular, the Sentinel-2 (S2) mission has great potential to produce reliable proxies for biological diversity. In beech forests of two Italian National Parks, we sampled the beetle fauna, breeding birds, and epiphytic lichens. First, we calculated Shannon’s entropy and Simpson’s diversity. Then, to produce variables for biodiversity assessment, we exploited S2 data acquired in the 4 years 2017–2021. S2 images were used to construct spectral bands and photosynthetic indices time series, from which 91 harmonic metrics were derived. For each taxon and multi-taxon community, we assessed the correlation with S2 harmonic metrics, biodiversity indices, and forest structural variables. Then, to assess the potential of the harmonic metrics in predicting species diversity in terms of Shannon’s and Simpson’s biodiversity indices, we also fit a random forests model between each diversity index and the best 10 harmonic metrics (in terms of absolute correlation, that is, the magnitude of the correlation) for each taxon. The models’ performance was evaluated via the relative root mean squared error (RMSE%). Overall, 241 beetle, 27 bird, and 59 lichen species were recorded. The diversity indices were higher for the multi-taxon community than for the single taxa. They were generally higher in the CVDA site than in GSML, except for the bird community. The highest correlation values between S2 data and biodiversity indices were recorded in CVDA for multi-taxon and beetle communities (| r| = 0.52 and 0.38, respectively), and in GSML for lichen and beetle communities (| r| = 0.34 and 0.26, respectively). RMSE% ranged between 2.53 and 9.99, and between 8.1 and 16.8 for the Simpson and Shannon index, respectively. The most important variables are phase and RMSE of red-Edge bands for bird and lichen communities, while RMSE and time of tassel cap and from EVI indices for beetles and multi-taxon diversity. Our results demonstrate that S2 data can be used for identifying potential biodiversity hotspots, showing that the herein presented harmonic metrics are informative for several taxa inhabiting wood, giving concrete support to cost-effective biodiversity monitoring and nature-based forest management in complex mountain systems.

Some Results on Harmonic Metrics

In this paper, we prove that the harmonicity of a metric on a pseudo-Riemannian manifold is equivalent to the harmonicity of both its Sasaki (resp. complete) lift metric on the tangent bundle and its Sasaki lift metric (resp. the Riemannian extension of the Levi-Civita connection) on the cotangent bundle, meanwhile we deal with harmonic metrics on warped product spaces. Some applications are included.

Generalized Kantowski–Sachs type spacetime metrics and their harmonicity

In this paper, we deal with harmonic metrics with respect to generalized Kantowski–Sachs type spacetime metrics. We also consider the Sasaki, horizontal and complete lifts of generalized Kantowski–Sachs type spacetime metrics to tangent bundle and study their harmonicity.

Simpson–Mochizuki correspondence for λ-flat bundles

The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi–Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable λ-flat bundles (λ≠0). On the other hand, we study two applications of the Simpson–Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable λ-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behavior.

Asymptotic geometry of the moduli space of parabolic SL(2,C)$\operatorname{SL}(2,\mathbb {C})$‐Higgs bundles

Given a generic stable strongly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundle ( E , φ ) $({\mathcal {E}}, \varphi )$ , we describe the family of harmonic metrics h t $h_t$ for the ray of Higgs bundles ( E , t φ ) $({\mathcal {E}}, t \varphi )$ for t ≫ 0 $t\gg 0$ by perturbing from an explicitly constructed family of approximate solutions h t app $h_t^{\mathrm{app}}$ . We then describe the natural hyperkähler metric on M $\mathcal {M}$ by comparing it to a simpler ‘semiflat’ hyperkähler metric. We prove that g L 2 − g sf = O ( e − γ t ) $g_{L^2} \,{-}\, g_{\mathrm{sf}} \,{=}\, O({\mathrm{e}}^{-\gamma t})$ along a generic ray, proving a version of Gaiotto–Moore–Neitzke's conjecture. Our results extend to weakly parabolic SL ( 2 , C ) $\operatorname{SL}(2,\mathbb {C})$ -Higgs bundles as well. A centerpiece of this paper is our explicit description of the moduli space and its L 2 $L^2$ metric for the case of the four-punctured sphere. We prove that the hyperkähler manifold in this case is a gravitational instanton of type ALG and that its rate of exponential decay to the semiflat metric is the conjectured optimal one, γ = 4 L $\gamma =4L$ , where L $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric | det φ | $|\det \varphi |$ .

On the correlation of temperature harmonic content with energy dissipation in C45 steel samples under fatigue loading

Fatigue damage onset involves a transformation of mechanical energy. This is in part spent into work of plastic deformation and fracture and in part transformed into heat. Therefore, different thermomechanical heat sources are activated during a fatigue loading cycle, resulting in a temperature modulation that can be characterized by its frequency content. Several studies have monitored the temperature on samples undergoing high-cycle fatigue, measuring specific thermomechanical metrics which could serve as indicators of damage evolution. The present work evaluates the harmonic content of temperature, to investigate its correlation with the material dissipation. The harmonic terms investigated are the amplitude and phase of temperature at the loading frequency (First Harmonic) and at twice the loading frequency (Second Harmonic). The paper reviews the mechanisms linking the evolution of heat dissipation to changes in these four parameters. The theoretical analysis is supported by an experimental campaign carried out on C45 steel tensile specimen subject to R=−1 and R=0.1 loading ratios. The harmonic metrics are evaluated analyzing both their spatially averaged value and their spatial dispersion, to check for the formation of localized heat dissipation zones in early fatigue life. The study shows that the phase of the Second Harmonic signal has a peculiar bimodal distribution that can be correlated to either a thermoelastic or dissipative prevailing effect. This behavior evolves towards a unimodal dissipative behavior when the load amplitude is increased, thus proposing the Second Harmonic phase parameter as an effective indicator for monitoring internal intrinsic dissipation in the material.

Harmonic performance analysis of a wind driven micro grid inverter

ABSTRACT The paper envisages measures to reduce the total harmonic distortion (THD) and harmonic spread factor (HSF) of the output voltage of a micro grid inverter. It involves the design of a variable structure controller (VSC) to derive the pulses for operating the switches in the inverter and there from enable the shaping of the output voltage to a nearly sinusoidal form. The methodology originates from the state space model of the system and brings in the control law to govern the operation of the PWM scheme. The framework includes a front end converter to exploit the maximum available power and engages a two stage process to provide a variable ac output to the utilities in the stand alone grid. The MATLAB based simulation results exhibit the performance improvement using the harmonic metrics to regulated output voltage over the range of operating loads and also the prototype registers to validate the simulated response.

Comparison of Multi-Temporal PlanetScope Data with Landsat 8 and Sentinel-2 Data for Estimating Airborne LiDAR Derived Canopy Height in Temperate Forests

Developing accurate methods for estimating forest structures is essential for efficient forest management. The high spatial and temporal resolution data acquired by CubeSat satellites have desirable characteristics for mapping large-scale forest structural attributes. However, most studies have used a median composite or single image for analyses. The multi-temporal use of CubeSat data may improve prediction accuracy. This study evaluates the capabilities of PlanetScope CubeSat data to estimate canopy height derived from airborne Light Detection and Ranging (LiDAR) by comparing estimates using Sentinel-2 and Landsat 8 data. Random forest (RF) models using a single composite, multi-seasonal composites, and time-series data were investigated at different spatial resolutions of 3, 10, 20, and 30 m. The highest prediction accuracy was obtained by the PlanetScope multi-seasonal composites at 3 m (relative root mean squared error: 51.3%) and Sentinel-2 multi-seasonal composites at the other spatial resolutions (40.5%, 35.2%, and 34.2% for 10, 20, and 30 m, respectively). The results show that RF models using multi-seasonal composites are 1.4% more accurate than those using harmonic metrics from time-series data in the median. PlanetScope is recommended for canopy height mapping at finer spatial resolutions. However, the unique characteristics of PlanetScope data in a spatial and temporal context should be further investigated for operational forest monitoring.

Mapping Forest Composition with Landsat Time Series: An Evaluation of Seasonal Composites and Harmonic Regression

The Landsat program has long supported pioneering research on the recovery of forest information by remote sensing technologies for several decades, and efforts to improve the thematic resolution and accuracy of forest compositional products remains an area of continued innovation. Recent development and application of Landsat time series analysis offers unique opportunities for quantifying seasonality and trend components among different forest types for developing alternative feature sets for forest vegetation mapping. Within a large forested landscape in Southeastern Ohio, USA, we examined the use of harmonic metrics developed from time series of all available Landsat-8 observations (2013–2019) relative to seasonal image composites, including accompanying spectral components and vegetation indices. A reference dataset among three sources was integrated and used to categorize forest inventory data into seven forest type classes and gradient compositional response. Results showed that the combination of harmonic metrics and topographic variables achieved an accuracy agreement with the reference data of 74.9% relative to seasonal composites (71.6%) and spectral indices (70.3%). Differences in agreement were attributed to improved discrimination of three heterogeneous upland hardwood classes and an early-successional, young forest class, all forest types of primary interest among managers across the region. Variable importance metrics often identified the cosine and sine terms that quantify the seasonality in spectral values in the harmonic feature space, suggesting these aspects best support the characterization of forest types at greater thematic detail than seasonal compositing procedures. This study demonstrates how advanced time series metrics can improve forest type modeling and forest gradient quantifications, thus showcasing a need for continued exploration of such approaches across different forest types.

Harmonic metrics on Higgs sheaves and uniformization of varieties of general type

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-étale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one. As a further application, we establish a nonabelian Hodge correspondence on smooth loci of klt varieties.

Improved mapping of forest type using spectral-temporal Landsat features

Multi-spectral imagery from the Landsat family of satellites has been used to map forest properties for decades, but accurate forest type characterizations at a 30-m Landsat resolution have remained an ongoing challenge, especially over large areas. We combined existing Landsat time series algorithms to quantify both harmonic and phenological metrics in a new set of spectral-temporal features that can be produced seamlessly across many Landsat scenes. Harmonic metrics characterize mean annual reflectance and seasonal variability, while phenological metrics quantify the timing of seasonal events. We assessed the performance of spectral-temporal features derived from time series of all available observations (1985–2015) relative to more conventional single date and multi-date inputs. Performance was determined based on agreement with a reference dataset for eight New England forest types at both the pixel and polygon scale. We found that spectral-temporal features consistently and significantly (paired t-test, p ≪ 0.01) outperformed all feature sets derived from individual images and multi-date combinations in all measures of agreement considered. Harmonic features, such as annual amplitude and model fit error, aid in distinguishing deciduous hardwoods from conifer species, while phenology features, like the timing of autumn onset and growing season length, were useful in separating hardwood classes. This study represents an important step toward large-scale forest type mapping using spectral-temporal Landsat features by providing a quantitative assessment of the advantages of harmonic and phenology features derived from time series of Landsat data as compared with more conventional single-date and multi-date classification inputs.

English

We study harmonic metrics with respect to the class of invariant metrics on non-reductive homogeneous four dimensional manifolds. In particular, we consider harmonic lifted metrics with respect to the Sasaki lifts, horizontal lifts and complete lifts of the metrics under study.

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces

Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree $0$ on a compact connected Riemann surface. Once we fix the flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t$ $(t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,t\theta)$ such that $\det(h_t)=h_{\det(E)}$. We study the behaviour of $h_t$ when $t$ goes to $\infty$. First, we show that the Hitchin equation is asymptotically decoupled under the assumption that the Higgs field is generically regular semisimple. We apply it to the study of the so called Hitchin WKB-problem. Second, we study the convergence of the sequence $(E,\overline{\partial}_E,\theta,h_t)$ in the case where the rank of $E$ is two. We introduce a rule to determine the parabolic weights of a "limiting configuration", and we show the convergence of the sequence to the limiting configuration in an appropriate sense. The results can be appropriately generalized in the context of Higgs bundles with a Hermitian-Einstein metric on curves.

Meromorphic differentials with twisted coefficients on compact Riemann surfaces

We apply the Dirichlet’s principle to a modified energy functional on Riemann surfaces to reprove the existence of harmonic metrics with certain prescribed singularities due to Simpson, Sabbah and Biquard–Boalch, and hence of differentials with twisted coefficients of the second and third kinds. As a by-product, this generalizes the classical theory of Abelian differentials on a compact Riemann surface to the case of twisted coefficients. This also proposes a more natural approach for general existence of harmonic metrics in the higher dimensional case.

On the biharmonicity of product maps

We introduce the warped product of maps defined between Riemannian warped product spaces and we give necessary and sufficient conditions for warped product maps to be (bi)harmonic. We obtain then some characterizations of nontrivial harmonic metrics and nonharmonic biharmonic metrics on warped product spaces.

Degenerations of mixed Hodge structure

We extend certain aspects of C. Simpson's correspondence between harmonic metrics and variations of Hodge structure to the category of complex variations of mixed Hodge structure, and we prove an analog of W. Schmid's nilpotent orbit theorem for admissible variations of graded-polarized mixed Hodge structure $\mathscr {V} \Delta\sp \ast$.

Degenerations of mixed Hodge structure

We extend certain aspects of C. Simpson's correspondence between harmonic metrics and variations of Hodge structure to the category of complex variations of mixed Hodge structure, and we prove an analog of W. Schmid's nilpotent orbit theorem for admissible variations of graded-polarized mixed Hodge structure $\mathscr {V} \Delta\sp \ast$.