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If f: U R2!R is a solution of the minimal surface equation, then for all nonnegative Lipschitz functions : R3!R with support contained in U R, Z graph(f) jAj2 2d˙ C Z graph(f) jr graph(f) j 2d˙ /Length 3024 43o ��lʮ��OU�-@6�]U�hj[��2�M�uW�Ũ� ^�t��n�Au��|��x�#*P�,i��˘��. On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain. And we will study when the sharp isoperimetric inequality for the minimal surface follows from that of the two flat surfaces. Anal.45, 194–221 (1972), Lawson, Jr., B.: Lectures on Minimal Submanifolds. It is well-known that a minimal graph of codimension one is stable, i.e. PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of minimal surfaces and eigenvalues of the laplacian. $\begingroup$ The problem asks for the stability of the minimal surface. Barbosa, J. L. (et al.) Brasil. Classify minimal surfaces in R3 whose Gauss map is one to one (see Theorem 9:4 in Osserman’s book). Annals of Mathematics Studies21, Princeton: Princeton, University Press 1951, Simons, J.: Minimal Varieties in Riemannian manifolds. The inequality was used by Simon in [Si] to show, among other things, that stable minimal hypercones of R n + 1 must be planar for n ≤ 6 and it was subsequently used to infer curvature estimates for stable minimal hypersurfaces, generalizing the classical work of Heinz [He], cf. We note that the construction of the index in this space (in the sense of Fischer-Colbrie [FC85]) in Section 4 is somewhat subtle. On the size of a stable minimal surface in R 3. %PDF-1.5 J. The proof of Theorem 1.2 uses crucially the fact that for two-dimensional minimal surfaces the sum of the squares of the principal curvatures 2 1 + 2 2 equals 2 1 2 = 2K, where Kis the Gauˇ curvature |since on a minimal surface 1 + 2 = 0. Subscription will auto renew annually. A complex version of the stability inequality for minimal surfaces was derived, in-cluding curvature terms for the case of an underlying space which is not at. Math. The case involving both charge and angular momentum has been proved recently in [25]. Let S be a stable minimal surface. So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. 3 Gauss curvature for stable minimal surfaces in R3, which yielded the Bern-stein theorem for complete stable minimal surfaces in R3. For basics of hypersurface geometry and the derivation of the stability inequality, Simons’ identity and the Sobolev inequality on minimal hypersurfaces, [S] is an excellent reference. For the integral estimates on jAj, follow the paper [SSY]. Marginally trapped surfaces are of central importance in general relativity, where they play the role of apparent horizons, or quasilocal black hole bound-aries. We can run the whole minimal model program for the moduli space of Gieseker stable sheaves on P2 via wall crossing in the space of stability conditions. The stability inequality can be used to get upper bounds for the total curvature in terms of the area of a minimal surface. the second variation of the area functional is non-negative. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is ﬂat. 68 0 obj Learn more about Institutional subscriptions, Barbosa, J.L., do Carmo, M.: On the size of a stable minimal surface inR Then, take f = 1 in the stability inequality Q (f) 0 to nd jIIj2 + Ric g( ; ) d 0: Because jIIj2 0 and Ric g( ; ) >0 by assumption, this is a contradiction. Remarks. Publication: Abstract and Applied Analysis. stream Many papers have been devoted to investigating stability. At the same time, Fischer-Colbrie and Schoen [12], independently, showed minimal surfaces: Corollary 2. << Pure Appl. In particular, we consider the space of so-called stable minimal surfaces. n An. Math. ... A theorem of Hopf and the Cauchy-Riemann inequality. A minimal surface is called stable if (and only if) the second variation of the area functional is nonnegative for all compactly supported deformations. A strong stability condition on minimal submanifolds Chung-Jun Tsai National Taiwan University Abstract: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. xڵ�r۸�=_�>U��:��N�u'��&޶3�}�%��$zI�^�}��)��l'm��@>��^�'��x��{�gB�\k9��L0��r��]�f?�8��7N�s��? /Filter /FlateDecode By plugging a … Processing of Telemetry Data Generated By Sensors Moving in a Varying Field (M113) D.J. Proof. The slope inequality asserts that ω2 f ≥ 4g −4 g deg(f∗ωf) for a relative minimal ﬁbration of genus g ≥ 2. Rend. If (M;g) has positive Ricci curvature, then cannot be stable. © 2021 Springer Nature Switzerland AG. Math. Comm. We do not know the smallest value of a for which A-aK has a positive solution. Destination page number Search scope Search Text Search scope Search Text Index, vision number and stability of complete minimal surfaces. The operators A - aK are intimately connected with the stability of minimal surfaces, the case a = 2 for surfaces in R3, and the case Q = 1 for surfaces in scalar flat 3-manifolds (see Theorem 4). On the Size of a Stable Minimal Surface in R 3 Pages 115-128. Arch. Assume that is stable. The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true The key underlying property of the local versions of the inequality is the notion of stability, both for minimal hypersurfaces and for … $\endgroup$ – User4966 Nov 21 '14 at 7:12 We note that a noncompact minimal surface is said to be stable if its index is zero. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ(K S ) ( [16]). For the minimal surface problem associated with (6) – (7), it is shown in Section 4 of Chapter … Moreover, the minimal model is smooth. Ann. J. In chemical reactions involving a solid material, the surface area to volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed. Mech.14, 1049–1056 (1965), Spruck, J.: Remarks on the stability of minimal submanifolds ofR Scand.5, 15–20 (1957), Polya, G., Szegö, G.: Isoperimetric inequalities of Mathematical Physics. We link these stability properties with the surface gravity of the horizon and/or to the existence of minimal sections. of Math.40, 834–854 (1939), Smale, S.: On the Morse index theorem. [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. Received 15 September 1979; First Online 01 February 2012; DOI https://doi.org/10.1007/978-3-642-25588-5_15 }z"��9Qr~��3M��-��ٛo>��O�� y6��ӻ�_.�>�3��l!˳p��W�E�7=��n��H��k��|��9s��瀠Rj~��Ƿ?�`�{/"�BgLh=[(˴�h�źlK؛��A��|{"��ƛr�훰bWweKë� �h��Uq"-�ŗm��z��'\W��܅��-�y@�v�ݖ�g��(��K��O�7D:B�,@�:��zG�vYl��}s{�3�B��݊��l� �)7EW�VQ��îm��]y��Wz:xLp��EV��+|Z@#�_ʦ��G\��8s��H�� C�V��뀯�ՉC�I9_��):حK�~U5mGC��)O�|Y��~S'�̻�s�=�֢I�S��S��R��D�eƸ�=� ��8�H8�Sx0>�`�:Y��Y0� ��޵ժDE��["m��x�V� In particular, F(E) F(K) = njKj whenever jEj= jKj. n. Math. Z.162, 245–261 (1978), Barbosa, J.L., do Carmo, M.: A necessary condition for a metric inR For an immersed minimal surface in R3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. A Stability Inequality For a Class of Nonlinear Feedback Systems (M114) A.G. Dewey. References 3. § are C1, parameterized by arclength, such that the tangent vector t = c0 is absolutely continuous. A complex version of the stability inequality for minimal surfaces was derived, in-cluding curvature terms for the case of an underlying space which is not at. Barbosa and do Carmo showed [9] that an orientable minimal surface in R3 for which the area of the im-age of the Gauss map is less than 2 π is stable. : Complete minimal surfaces with total curvature −2π. Comment. We establish the Noether inequality for projective 3-folds, and, specifically, we prove that the inequality vol (X) ≥ 4 3 p g (X) − 10 3 holds for all projective 3-folds X of general type with either p g (X) ≤ 4 or p g (X) ≥ 21, where p g (X) is the geometric genus and vol (X) is the canonical volume. a time symmetric Cauchy surface, then θ+ = 0 if and only if Σ is minimal. Indeed, the role of … Nashed, M.Zuhair; Scherzer, Otmar. This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. Let M be a minimal surface in the simply-connected space form of constant curvature a, and let D be a simply-connected compact domain with piecewise smooth boundary on M. Let A denote the second fundamental form of M . Arch. strict stability of , we prove that a neighborhood of it in Mis iso- ... of a stable minimal surface ˆMwas in the proof of the positive mass theorem given by Schoen and Yau [17]. Then A 4πQ2 , (43) where A is the area of S and Q is its charge. at the pointwise estimate. The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. Jber. In recent years, a lot of attention has been given to the stability of the isoperimetric/Wul inequality. The Gauss-Bonnet Theorem (see Singer and Thorpe’s book). Preprint, Chern, S.S.: Minimal submanifolds in a Riemannian manifold. A stability criterion can be seen as a set of inequality constraints describing the conditions under which these equalities are preserved. uis minimal. The conjectured Penrose inequality, proved in the Riemannian case by Pages 167-182. The main goal of this article is to extend this result in several directions. A theorem of Micallef, which makes use of the complex stability inequality, states that any complete parabolic two-dimensional surface in four-dimensional Euclidean space is holomorphic Math. 2. Barbosa, João Lucas (et al.) It was Severi who stated it as a theorem in [Se], whose proof was not correct unfortunately. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. The Sobolev inequality (see Chapter 3). (joint with R. Schoen) Mar 28, 2019 (Thur) 11:00-12:00 @ AB1 502a (Note special date and time.) The Sobolev inequality (see Chapter 3). Deﬁnition 2. (to appear), Bandle, C.: Konstruktion isomperimetrischer Ungleichungen der Mathematischen Physik aus solchen der Geometrie. Palermo33 201–211 (1912), Nitsche, J.: A new uniqueness theorem for minimal surfaces. In [10] do Carmo and Peng gave https://doi.org/10.1007/BF01215521, Over 10 million scientific documents at your fingertips, Not logged in Amer. can get a stability-free proof of the slope inequality. In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV ... establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. Springer, Berlin, Heidelberg. Mat. To learn the Moser iteration technique, follow [GT]. Stable Approximations of a Minimal Surface Problem with Variational Inequalities M. Zuhair Nashed 1 and Otmar Scherzer 2 1 Department of Mathematical Sciences, University of … volume 173, pages13–28(1980)Cite this article. 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. I.M.P.A., Rio de Janeiro: Instituto de Matematica Pura e Applicada 1973, Lichtenstein, L.: Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung von elliptischem typus. Rational Mech. Acad. TheDirichlet problem forthe minimal surface problem istoﬁndafunction u of minimal area A(u), as deﬁned in (6) – (7), in the class BV(Ω) with prescribeddataφon∂Ω. J. Math.98, 515–528 (1976), Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. inequality to higher codimension, to non local perimeters and to non euclidean settings such as the Gauss space. 3 Stable minimal surfaces and the ﬁrst eigenvalue The purpose of this section is to obtain upper bounds for the ﬁrst eigenvalue of stable minimal surfaces, which are deﬁned as follows. 162, … Stable approximations of a minimal surface problem with variational inequalities Nashed, M. Zuhair; Scherzer, Otmar; Abstract. %�� Deutsch. If rankL = 1 or 2 then x(M) is a quotient of the plane, the helicoid or a Scherk's surface. Stability of Minimal Surfaces and Eigenvalues of the Laplacian. This inequality … Jury. >> Let X be a smooth minimal surface of general type over kand of maximal Albanese dimension. Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. Stability inequality : Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included : S.02.a: Bernstein problem : Daniel Paunovic [CM11] conclusion of Ch.1 §5 : S.02.b: 28.10. Math.-Verein.51, 219–257 (1941), Chen, C.C. interval (δ, 1], where δ > 0, and the extrinsic curvature of the surface satisfies the inequality \K e \ > 3(1 - δ)2/2δ. Math. Finally, Section 6 gives an account on how the techniques developed for the isoperimetric inequality have been successfully applied to study the stability of other related inequalities. (i) The maximal quotients of the helicoid and the Scherk's surfaces … https://doi.org/10.1007/978-3-642-25588-5_15. The Zero-Moment Point (ZMP) [1] criterion, namely that The Wul inequality states that, for any set of nite perimeter EˆRn, one has F(E) njKj1n jEj n 1 n; (1.1) see e.g. Tax calculation will be finalised during checkout. This is a preview of subscription content, access via your institution. Math Z 173, 13–28 (1980). Amer. Barbosa J.L., Carmo M.. (2012) Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Mathematische Zeitschrift It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. More precisely, a minimal surface is stable if there are no directions which can decrease the area; thus, it is a critical point with Morse index zero. Ci. Exercise 6. Pogorelov [22]). Curves with weakly bounded curvature Let § be 2-manifold of class C2. On the other hand, we can use either the Gauss–Bonnet theorem or the Jacobi equation to get the opposite bound. Speaker: Chao Xia (Xiamen University) Title: Stability on … 1See [CM1] [CM2] for further reference. n+1 to be isometrically and minimally immersed inM Rational Mech. 1 In [16] the expected inequality for area and charge has been proved for stable minimal surfaces on time symmetric initial data. [17, 15]. ... J. Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, MSRI preprint. It is the curvature characteristic of minimal surfaces that is important. Hence our theorem can be regarded as an extension of the results in [ 6 – 8 ]. A Reverse Isoperimetric Inequality and Extremal Theorems 3 1. minimal surface. In §5 we prove a theorem on the stability of a minimal surface in R4, which does not have an analogue for 3-dimensional spaces. Anal.58, 285–307 (1975), Peetre, J.: A generalization of Courant's nodal domain theorem. Anal.52, 319–329 (1973), Morrey, Jr., C.B., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. outermost minimal surface is a minimal surface which is not contained entirely inside another minimal surface. In fact, all the known proofs are related to some sort of stability: geometric invariant theory in [CH] (in Stable approximations of a minimal surface problem with variational inequalities. 98, 515–528 (1976) Google Scholar. Minimal surfaces of small total curvature : Martina Jorgensen The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). We identify a strong stability condition on minimal submanifolds that generalizes the above scenario. The surface-area-to-volume ratio, also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. First, we prove the inequality for generic dynamical black holes. For the … Rational Mech. Math.10, 271–290 (1957), Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. Arch. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. the link-to-surface distance) while a ﬁxed contact constraints all six DOFs of the end-effector link. Using the inequality of the Lemma for m = 2, we can improve the stability theorem of Barbosa and do Carmo [2]. Stability of surface contacts for humanoid robots: ... issue, as its dimension is minimal (six). Jaigyoung Choe's main interest is in differential geometry. ;�0,3�r˅+��,cJ�"MbF��b��B;�N�*��? Abstract and Applied Analysis (1997) Volume: 2, Issue: 1-2, page 137-161; ISSN: 1085-3375; Access Full Article top Access to full text Full (PDF) How to cite top His key idea was to apply the stability inequality[See §1.2] to diﬀerent well chosen functions. It is obvious that a complete stable minimal hypersurface in $\mathbb{H}^{n+1}(-1)$ has index 0. It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. A classi ca-tion theorem for complete stable minimal surfaces in three-dimensional Riemannian manifolds of nonnegative scalar curvature has been obtained by Fischer-Colbrie and Schoen [3].

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