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Journal articles on the topic 'Scalar curvature problem'

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Author: Grafiati

Published: 14 March 2023

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1

BUCATARU, IOAN, and ZOLTÁN MUZSNAY. "FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM." Journal of the Australian Mathematical Society 97, no. 1 (May 20, 2014): 27–47. http://dx.doi.org/10.1017/s1446788714000111.

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AbstractIt is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these conditions the scalar flag curvature (SFC) test. The proof of the main result provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. Hilbert’s fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert’s fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the constant flag curvature (CFC) test, which we developed in our previous paper, Bucataru and Muzsnay [‘Sprays metrizable by Finsler functions of constant flag curvature’, Differential Geom. Appl.31 (3)(2013), 405–415] as well as the SFC test to decide whether or not the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the CFC and SFC tests to construct solutions to Hilbert’s fourth problem.

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2

Li, Ying, Xiaohuan Mo, and Yaoyong Yu. "Inverse problem of sprays with scalar curvature." International Journal of Mathematics 30, no. 09 (August 2019): 1950041. http://dx.doi.org/10.1142/s0129167x19500411.

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Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray [Formula: see text] on a manifold is of vanishing [Formula: see text]-curvature, but [Formula: see text] has not isotropic curvature, then [Formula: see text] is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain [Formula: see text] with scalar curvature and vanishing [Formula: see text]-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

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3

Cheng, Qing-Ming, Shichang Shu, and Young Jin Suh. "Compact hypersurfaces in a unit sphere." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 6 (December 2005): 1129–37. http://dx.doi.org/10.1017/s0308210500004303.

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We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

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4

Cheng, Xinyue, Li Yin, and Tingting Li. "A class of Randers metrics of scalar flag curvature." International Journal of Mathematics 31, no. 13 (November 18, 2020): 2050114. http://dx.doi.org/10.1142/s0129167x20501141.

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One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that [Formula: see text] is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

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5

Chen, Xuezhang, and Liming Sun. "Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds." Communications in Contemporary Mathematics 21, no. 03 (May 2019): 1850021. http://dx.doi.org/10.1142/s0219199718500219.

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We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

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6

Holcman, David. "Prescribed scalar curvature problem on complete manifolds." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 328, no. 4 (February 1999): 321–26. http://dx.doi.org/10.1016/s0764-4442(99)80218-3.

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7

Holcman, David. "Prescribed scalar curvature problem on complete manifolds." Journal de Mathématiques Pures et Appliquées 80, no. 2 (March 2001): 223–44. http://dx.doi.org/10.1016/s0021-7824(00)01181-8.

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8

Kendall, P. C., P. N. Robson, and J. E. Sitch. "Rib waveguide curvature loss: the scalar problem." IEE Proceedings J Optoelectronics 132, no. 2 (1985): 140. http://dx.doi.org/10.1049/ip-j.1985.0028.

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9

YANG, KWANG-WU. "ON WARPED PRODUCT MANIFOLDS - CONFORMAL FLATNESS AND CONSTANT SCALAR CURVATURE PROBLEM." Tamkang Journal of Mathematics 29, no. 3 (September 1, 1998): 203–21. http://dx.doi.org/10.5556/j.tkjm.29.1998.4272.

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In this paper, we study some geometric properties on doubly or singly warped product manifolds. In general, on a fixed topological product manifold, the problem for finding warped-product metrics satisfying certain curvature conditions are finally reduced to find positive solutions of linear or non-linear differential equations. Here, we are mainly interested in the following problems on essentially warped-product manifolds: one is the sufficient and necessary conditions for conformal flatness, and the other is to find warped-product metrics so that their scalar curvatures are contants.

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10

Chen, Bin, and Lili Zhao. "On a Yamabe Type Problem in Finsler Geometry." Canadian Mathematical Bulletin 60, no. 2 (June 1, 2017): 253–68. http://dx.doi.org/10.4153/cmb-2016-102-x.

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AbstractIn this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

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11

Veronelli, Giona. "Scalar Curvature via Local Extent." Analysis and Geometry in Metric Spaces 6, no. 1 (November 1, 2018): 146–64. http://dx.doi.org/10.1515/agms-2018-0008.

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AbstractWe give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.

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12

Chtioui, Hichem, Hichem Hajaiej, and Marwa Soula. "The scalar curvature problem on four-dimensional manifolds." Communications on Pure & Applied Analysis 19, no. 2 (2020): 723–46. http://dx.doi.org/10.3934/cpaa.2020034.

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13

Ben Ayed, Mohamed, Hichem Chtioui, and Mokhles Hammami. "The scalar-curvature problem on higher-dimensional spheres." Duke Mathematical Journal 93, no. 2 (June 1998): 379–424. http://dx.doi.org/10.1215/s0012-7094-98-09313-9.

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14

Wang, Zhi-Qiang, and Florin Catrina. "Symmetric solutions for the prescribed scalar curvature problem." Indiana University Mathematics Journal 49, no. 2 (2000): 0. http://dx.doi.org/10.1512/iumj.2000.49.1847.

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15

Ho, Pak Tung, and Seongtag Kim. "CR Nirenberg problem and zero Wester scalar curvature." Annals of Global Analysis and Geometry 58, no. 2 (June 22, 2020): 207–26. http://dx.doi.org/10.1007/s10455-020-09721-w.

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16

Ambrosetti, Antonio, and Andrea Malchiodi. "On the Symmetric Scalar Curvature Problem on Sn." Journal of Differential Equations 170, no. 1 (February 2001): 228–45. http://dx.doi.org/10.1006/jdeq.2000.3816.

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17

García, Gonzalo, and Jhovanny Muñoz. "About the Uniqueness of Conformal Metrics with Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundary." Revista de Ciencias 13 (September 4, 2011): 71–79. http://dx.doi.org/10.25100/rc.v13i0.644.

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Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

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18

HO, PAK TUNG. "RESULTS RELATED TO PRESCRIBING PSEUDO-HERMITIAN SCALAR CURVATURE." International Journal of Mathematics 24, no. 03 (March 2013): 1350020. http://dx.doi.org/10.1142/s0129167x13500201.

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In this paper, we consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M. Using geometric flow, we prove that for any negative smooth function f we can prescribe the pseudo-Hermitian scalar curvature to be f, provided that dim M = 3 and the CR Yamabe invariant of M is negative. On the other hand, we establish some uniqueness and non-uniqueness results on prescribing pseudo-Hermitian scalar curvature.

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19

Kawai, Shigeo. "Scalar curvatures of conformal metrics on Sn." Nagoya Mathematical Journal 140 (December 1995): 151–66. http://dx.doi.org/10.1017/s0027763000005468.

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In this paper we consider the following problem: Given a smooth function K on the n-dimensional unit sphere Sn(n ≥ 3) with its canonical metric g0, is it possible to find a pointwise conformal metric which has K as its scalar curvature? This problem was presented by J. L. Kazdan and F. W. Warner. The associated problem for Gaussian curvature in dimension 2 had been presented by L. Nirenberg several years before.

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20

Ghimenti, Marco G., and Anna Maria Micheletti. "Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary." Mathematical Biosciences and Engineering 30, no. 4 (2022): 1209–35. http://dx.doi.org/10.3934/era.2022064.

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<abstract><p>We study the stability of compactness of solutions for the Yamabe boundary problem on a compact Riemannian manifold with non umbilic boundary. We prove that the set of solutions of Yamabe boundary problem is a compact set when perturbing the mean curvature of the boundary from below and the scalar curvature with a function whose maximum is not too positive. In addition, we prove the counterpart of the stability result: there exists a blowing up sequence of solutions when we perturb the mean curvature from above or the mean curvature from below and the scalar curvature with a function with a large positive maximum.</p></abstract>

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21

Ben Mahmoud, Randa, and Hichem Chtioui. "Prescribing the scalar curvature problem on higher-dimensional manifolds." Discrete & Continuous Dynamical Systems - A 32, no. 5 (2012): 1857–79. http://dx.doi.org/10.3934/dcds.2012.32.1857.

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22

Ben Ayed, Mohamed, Yansong Chen, Hichem Chtioui, and Mokhles Hammami. "On the prescribed scalar curvature problem on $4$ -manifolds." Duke Mathematical Journal 84, no. 3 (September 1996): 633–77. http://dx.doi.org/10.1215/s0012-7094-96-08420-3.

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23

Ma, Li, and Hui Wang. "A minimization problem arising from prescribing scalar curvature functions." Mathematische Zeitschrift 222, no. 1 (May 1996): 1–6. http://dx.doi.org/10.1007/bf02621856.

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24

Zhang, DaQing. "The concentrated solutions on the nonhomogeneous scalar curvature problem." Journal of Mathematical Analysis and Applications 487, no. 1 (July 2020): 123967. http://dx.doi.org/10.1016/j.jmaa.2020.123967.

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25

Li, M., and Wang Hui. "A minimization problem arising from prescribing scalar curvature functions." Mathematische Zeitschrift 222, no. 1 (May 6, 1996): 1–6. http://dx.doi.org/10.1007/pl00004528.

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26

Sharaf, Khadijah Abdullah, and Hichem Chtioui. "Topological invariants for the scalar curvature problem on manifolds." Advances in Pure and Applied Mathematics 14, Special (March 2023): 31–47. http://dx.doi.org/10.21494/iste.op.2023.0936.

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27

Ambrosetti, Antonio, YanYan Li, and Andrea Malchiodi. "On the Yamabe problem and the scalar curvature problems under boundary conditions." Mathematische Annalen 322, no. 4 (April 1, 2002): 667–99. http://dx.doi.org/10.1007/s002080100267.

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28

COLEY, A. A. "SCALAR AVERAGING IN COSMOLOGY." International Journal of Modern Physics D 19, no. 14 (December 2010): 2361–64. http://dx.doi.org/10.1142/s0218271810018359.

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The averaging problem in cosmology is of considerable importance for the correct interpretation of cosmological data. In this essay an approach to averaging based on scalar curvature invariants is presented, which gives rise to significant effects on cosmological evolution.

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29

CHAKRAVARTY, GIRISH KUMAR, SUBHENDRA MOHANTY, and NAVEEN K. SINGH. "HIGGS INFLATION IN f(Φ, R) THEORY." International Journal of Modern Physics D 23, no. 04 (March 18, 2014): 1450029. http://dx.doi.org/10.1142/s0218271814500291.

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We generalize the scalar-curvature coupling model ξΦ2R of Higgs inflation to ξΦaRb to study inflation. We compute the amplitude and spectral index of curvature perturbations generated during inflation and fix the parameters of the model by comparing these with the Planck + WP data. We find that if the scalar self-coupling λ is in the range 10-5–0.1, parameter a in the range 2.3–3.6 and b in the range 0.77–0.22 at the Planck scale, one can have a viable inflation model even for ξ ≃ 1. The tensor to scalar ratio r in this model is small and our model with scalar-curvature couplings is not ruled out by observational limits on r unlike the pure [Formula: see text] theory. By requiring the curvature coupling parameter to be of order unity, we have evaded the problem of unitarity violation in scalar-graviton scatterings which plague the ξΦ2R Higgs inflation models. We conclude that the Higgs field may still be a good candidate for being the inflaton in the early universe if one considers higher-dimensional curvature coupling.

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30

Shi, Yuguang, Wenlong Wang, and Guodong Wei. "Total mean curvature of the boundary and nonnegative scalar curvature fill-ins." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 784 (January 23, 2022): 215–50. http://dx.doi.org/10.1515/crelle-2021-0072.

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Abstract In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, completely solving an open problem due to Gromov (see Question 1.1). Then we introduce a fill-in invariant (see Definition 1.2) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds via this fill-in invariant. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov’s two conjectures formulated in [M. Gromov, Four lectures on scalar curvature, preprint 2019] (see Conjecture 1.1 and Conjecture 1.2 below).

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31

Ma, Li. "On the existence of solutions of prescribing scalar curvature problem." Tsukuba Journal of Mathematics 24, no. 1 (June 2000): 133–37. http://dx.doi.org/10.21099/tkbjm/1496164050.

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32

Ndiaye, Cheikh Birahim. "Multiple Solutions for the Scalar Curvature Problem on the Sphere." Communications in Partial Differential Equations 31, no. 11 (November 2006): 1667–78. http://dx.doi.org/10.1080/03605300600635087.

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33

Bahri, A., and J. M. Coron. "The scalar-curvature problem on the standard three-dimensional sphere." Journal of Functional Analysis 95, no. 1 (January 1991): 106–72. http://dx.doi.org/10.1016/0022-1236(91)90026-2.

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34

Ben Ayed, M., K. El Mehdi, and M. Ould Ahmedou. "The scalar curvature problem on the four dimensional half sphere." Calculus of Variations and Partial Differential Equations 22, no. 4 (April 2004): 465–82. http://dx.doi.org/10.1007/s00526-004-0285-6.

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35

BEN AYED, Mohamed, and Habib FOURTI. "Scalar curvature type problem on the three dimensional bounded domain." Acta Mathematica Scientia 37, no. 1 (January 2017): 139–73. http://dx.doi.org/10.1016/s0252-9602(16)30122-9.

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36

LOSEV, A., and A. TURBINER. "MULTIDIMENSIONAL EXACTLY SOLVABLE PROBLEMS IN QUANTUM MECHANICS AND PULLBACKS OF AFFINE COORDINATES ON THE GRASSMANNIAN." International Journal of Modern Physics A 07, no. 07 (March 20, 1992): 1449–65. http://dx.doi.org/10.1142/s0217751x92000636.

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Multidimensional exactly solvable problems related to compact hidden-symmetry groups are discussed. Natural coordinates on homogeneous space are introduced. It is shown that a potential and scalar curvature of the problem considered have quite a simple form of quadratic polynomials in these coordinates. A mysterious relation between the potential and the curvature observed for SU(2) in Refs. 2 and 3 is obtained in a simple way.

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37

Yacoub, Ridha. "Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere 𝕊+3." Geometry 2014 (March 20, 2014): 1–9. http://dx.doi.org/10.1155/2014/582367.

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In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

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38

YAN, YU. "SOME COMPACTNESS RESULTS RELATED TO SCALAR CURVATURE DEFORMATION." Communications in Contemporary Mathematics 09, no. 01 (February 2007): 81–120. http://dx.doi.org/10.1142/s0219199707002356.

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Motivated by the prescribing scalar curvature problem, we study the equation [Formula: see text] on locally conformally flat manifolds (M,g) with R(g) ≡ 0. We prove that when K satisfies certain conditions and the dimension of M is 3 or 4, any positive solution u of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that on four-dimensional locally conformally flat scalar positive manifolds the solutions of [Formula: see text] can only have simple blow-up points.

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39

Fermi, Davide, Massimo Gengo, and Livio Pizzocchero. "On the Necessity of Phantom Fields for Solving the Horizon Problem in Scalar Cosmologies." Universe 5, no. 3 (March 11, 2019): 76. http://dx.doi.org/10.3390/universe5030076.

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We discuss the particle horizon problem in the framework of spatially homogeneous and isotropic scalar cosmologies. To this purpose we consider a Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime with possibly non-zero spatial sectional curvature (and arbitrary dimension), and assume that the content of the universe is a family of perfect fluids, plus a scalar field that can be a quintessence or a phantom (depending on the sign of the kinetic part in its action functional). We show that the occurrence of a particle horizon is unavoidable if the field is a quintessence, the spatial curvature is non-positive and the usual energy conditions are fulfilled by the perfect fluids. As a partial converse, we present three solvable models where a phantom is present in addition to a perfect fluid, and no particle horizon appears.

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40

Schwartz, Fernando A. "The Zero Scalar Curvature Yamabe problem on noncompact manifolds with boundary." Indiana University Mathematics Journal 55, no. 4 (2006): 1449–60. http://dx.doi.org/10.1512/iumj.2006.55.2733.

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41

Alghanemi, Azeb, Wael Abdelhedi, and Hichem Chtioui. "Prescribing the scalar curvature problem on the four-dimensional half sphere." Arabian Journal of Mathematics 6, no. 3 (October 12, 2016): 137–51. http://dx.doi.org/10.1007/s40065-016-0155-z.

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42

Chtioui, Hichem, Khalil El Mehdi, and Najoua Gamara. "The Webster scalar curvature problem on the three dimensional CR manifolds." Bulletin des Sciences Mathématiques 131, no. 4 (June 2007): 361–74. http://dx.doi.org/10.1016/j.bulsci.2006.05.003.

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43

Chtioui, Hichem. "The Webster scalar curvature problem on higher dimensional CR compact manifolds." Comptes Rendus Mathematique 345, no. 1 (July 2007): 11–13. http://dx.doi.org/10.1016/j.crma.2007.05.006.

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44

Ben Mahmoud, Randa, Hichem Chtioui, and Afef Rigane. "On the prescribed scalar curvature problem on : The degree zero case." Comptes Rendus Mathematique 350, no. 11-12 (June 2012): 583–86. http://dx.doi.org/10.1016/j.crma.2012.06.012.

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45

Matsuo, Shinichiroh. "The prescribed scalar curvature problem for metrics with unit total volume." Mathematische Annalen 360, no. 3-4 (May 9, 2014): 675–80. http://dx.doi.org/10.1007/s00208-014-1052-4.

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46

Wei, Juncheng, and Shusen Yan. "Infinitely many solutions for the prescribed scalar curvature problem on SN." Journal of Functional Analysis 258, no. 9 (May 2010): 3048–81. http://dx.doi.org/10.1016/j.jfa.2009.12.008.

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47

Li, Benling, and Zhongmin Shen. "Sprays of isotropic curvature." International Journal of Mathematics 29, no. 01 (January 2018): 1850003. http://dx.doi.org/10.1142/s0129167x18500039.

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In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity [Formula: see text] vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.

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48

Shojaee, Neda, and Morteza MirMohammad Rezaii. "On the conformal scalar curvature equations on Finsler manifolds." International Journal of Geometric Methods in Modern Physics 14, no. 01 (December 20, 2016): 1750008. http://dx.doi.org/10.1142/s0219887817500086.

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In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

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49

Catino, Giovanni, Filippo Gazzola, and Paolo Mastrolia. "A conformal Yamabe problem with potential on the Euclidean space." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 5 (February 10, 2021): 1987–98. http://dx.doi.org/10.1007/s10231-021-01067-9.

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AbstractWe consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

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50

Kong, De-Xing, Qi Liu, and Chang-Ming Song. "Classical solutions to a dissipative hyperbolic geometry flow in two space variables." Journal of Hyperbolic Differential Equations 16, no. 02 (June 2019): 223–43. http://dx.doi.org/10.1142/s0219891619500085.

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We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.

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