{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:36:02Z","timestamp":1760236562374,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,12,5]],"date-time":"2021-12-05T00:00:00Z","timestamp":1638662400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and K\u00e4hlerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.<\/jats:p>","DOI":"10.3390\/axioms10040333","type":"journal-article","created":{"date-parts":[[2021,12,5]],"date-time":"2021-12-05T20:59:33Z","timestamp":1638737973000},"page":"333","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0591-8307","authenticated-orcid":false,"given":"Vladimir","family":"Rovenski","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1734-8874","authenticated-orcid":false,"given":"Sergey","family":"Stepanov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9186-3992","authenticated-orcid":false,"given":"Irina","family":"Tsyganok","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yano, K., and Bochner, S. (1953). Curvature and Betti Numbers, Princeton University Press.","DOI":"10.1515\/9781400882205"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Besse, A.L. (1987). Einstein Manifolds, Springer.","DOI":"10.1007\/978-3-540-74311-8"},{"key":"ref_3","first-page":"259","article-title":"Op\u00e9rateur de courbure et laplacien des formes diff\u00e9rentielles d\u2019une vari\u00e9t\u00e9 riemannianne","volume":"54","author":"Gallot","year":"1975","journal-title":"J. Math. Pures. Appl."},{"key":"ref_4","unstructured":"Wu, H.-H. (2018). The Bochner Technique in Differential Geometry, Classical Topics in Mathematics, Higher Education Press."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Petersen, P. (2016). Riemannian Geometry, Springer AG. [3rd ed.].","DOI":"10.1007\/978-3-319-26654-1"},{"key":"ref_6","unstructured":"Yano, K. (1970). Integral Formulas in Riemannian Geometry, Marcel Dekker."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1090\/S0273-0979-1988-15679-0","article-title":"From vanishing theorems to estimating theorems: The Bochner technique revisited","volume":"19","author":"Berard","year":"1988","journal-title":"Bull. AMS"},{"key":"ref_8","unstructured":"Pigola, S., Rigoli, M., and Setti, A.G. (2008). Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique, Progress in Mathematics; Birkh\u00e4user Verlag AG."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pigola, S., Rigoli, M., and Setti, A.G. (2005). Maximum Principles on Riemannian Manifolds and Applications, AMS.","DOI":"10.1090\/memo\/0822"},{"key":"ref_10","unstructured":"Grigor\u2019yan, A. (2009). Heat Kernel and Analysis on Manifolds, AMS\/IP."},{"key":"ref_11","unstructured":"Schoen, R., and Yau, S.T. (1994). Lectures on Harmonic Maps, International Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s10455-020-09714-9","article-title":"An example Lichnerowicz-type Laplacian","volume":"58","author":"Rovenski","year":"2020","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.difgeo.2017.03.006","article-title":"Liouville-type theorems for some classes of Riemannian almost product manifolds and for special mappings of Riemannian manifolds","volume":"54","author":"Stepanov","year":"2017","journal-title":"Differ. Geom. Appl."},{"key":"ref_14","unstructured":"Narasimhan, R. (1968). Analysis on real and complex manifolds. Advanced Studies in Pure Mathematics, North-Holland Publishing Co."},{"key":"ref_15","first-page":"369","article-title":"Remarks on conformal transformations","volume":"8","author":"Yau","year":"1973","journal-title":"J. Diff. Geom."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Eisenhart, L.P. (1949). Riemannian Geometry, Princeton University Press.","DOI":"10.1515\/9781400884216"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1090\/S0273-0979-99-00776-4","article-title":"Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds","volume":"36","year":"1999","journal-title":"Bull. Amer. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1090\/S0002-9947-1992-1033232-2","article-title":"Superharmonic functions on foliations","volume":"330","author":"Adams","year":"1992","journal-title":"Trans. AMS"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"89","DOI":"10.4310\/jdg\/1214510047","article-title":"Vanishing theorems on complete Kahler manifolds and their applications","volume":"50","author":"Ni","year":"1998","journal-title":"J. Differ. Geom."},{"key":"ref_20","first-page":"191","article-title":"On the heat kernel of a complete Riemannian manifold","volume":"57","author":"Yau","year":"1978","journal-title":"J. Math. Pures Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01351342","article-title":"Non-existence of continuous convex functions on certain Riemannian manifolds","volume":"207","author":"Yau","year":"1974","journal-title":"Math. Ann."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1512\/iumj.1976.25.25051","article-title":"Erratum: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry","volume":"25","author":"Yau","year":"1976","journal-title":"Indiana Univ. Math. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s00032-008-0084-1","article-title":"Aspect of potential theory on manifolds, linear and non-linear","volume":"76","author":"Pigola","year":"2008","journal-title":"Milan J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9947-1969-0251664-4","article-title":"Manifolds of negative curvature","volume":"145","author":"Bishop","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Rovenski, V. (1998). Foliations on Riemannian Manifolds and Submanifolds, Birkh\u00e4user Boston.","DOI":"10.1007\/978-1-4612-4270-3_1"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1216\/rmjm\/1181072105","article-title":"A survey on paracomplex geometry","volume":"26","author":"Cruceanu","year":"1996","journal-title":"Rocky Mountain J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1134\/S0037446620040102","article-title":"The bundle of paracomplex structures","volume":"61","author":"Kornev","year":"2020","journal-title":"Sib. Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1070\/RM2009v064n01ABEH004591","article-title":"Homogeneous para-K\u00e4hlerian Einstein manifolds","volume":"64","author":"Medori","year":"2009","journal-title":"Russian Math. Surv."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1016\/j.geomphys.2014.09.003","article-title":"On solutions to equations with partial Ricci curvature","volume":"86","author":"Rovenski","year":"2014","journal-title":"J. Geom. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s11401-014-0871-7","article-title":"Bochner-Kodaira techniques on K\u00e4hler Finsler manifolds","volume":"36","author":"Xiao","year":"2015","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3047","DOI":"10.1016\/S0362-546X(01)00424-2","article-title":"The introduction of Bochner\u2019s technique on Lorentzian manifolds","volume":"47","author":"Romero","year":"2001","journal-title":"Nonlinear Anal."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1007\/s10958-007-0024-6","article-title":"Vanishing theorems in affine, Riemann and Lorentzian geometries","volume":"141","author":"Stepanov","year":"2007","journal-title":"J. Math. Sci. (N. Y.)"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s00222-020-01003-3","article-title":"New curvature conditions for the Bochner technique","volume":"224","author":"Petersen","year":"2021","journal-title":"Invent. Math."},{"key":"ref_34","unstructured":"Schoen, R., and Yau, S.-T. (2010). Lectures on Differential Geometry, International Press of Boston."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Topping, P. (2006). Lectures on the Ricci Flow, Cambridge University Press.","DOI":"10.1017\/CBO9780511721465"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/333\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:39:53Z","timestamp":1760168393000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/333"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,5]]},"references-count":35,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["axioms10040333"],"URL":"https:\/\/doi.org\/10.3390\/axioms10040333","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,12,5]]}}}