stability inequality minimal surface

Mathematische Zeitschrift A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ (K S) (). Processing of Telemetry Data Generated By Sensors Moving in a Varying Field (M113) D.J. This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. Pages 167-182. minimal surface in hyperbolic space satisfy the following relation (Gauss’ lemma): K = −1− |B|2 2. For the systems that concern us in subsequent chapters, this area property is irrelevant. 1See [CM1] [CM2] for further reference. 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. Hence our theorem can be regarded as an extension of the results in [ 6 – 8 ]. It was Severi who stated it as a theorem in [Se], whose proof was not correct unfortunately. The J. Analyse Math.19, 15–34 (1967), Kaul, H.: Isoperimetrische Ungleichung und Gauss-Bonnet-Formel fürH-Flächen in Riemannschen Mannigfaltigkeiten. 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. Mat. His key idea was to apply the stability inequality[See §1.2] to diﬀerent well chosen functions. Z.162, 245–261 (1978), Barbosa, J.L., do Carmo, M.: A necessary condition for a metric inR For the integral estimates on jAj, follow the paper [SSY]. In recent years, a lot of attention has been given to the stability of the isoperimetric/Wul inequality. Nashed, M.Zuhair; Scherzer, Otmar. 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. 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 . Tax calculation will be finalised during checkout. n. Math. 2 J. ... A theorem of Hopf and the Cauchy-Riemann inequality. 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 … If rankL = 1 or 2 then x(M) is a quotient of the plane, the helicoid or a Scherk's surface. A Reverse Isoperimetric Inequality and Extremal Theorems 3 1. [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. Amer. Lemma. Destination page number Search scope Search Text Search scope Search Text © 2021 Springer Nature Switzerland AG. Curves with weakly bounded curvature Let § be 2-manifold of class C2. If the free-surface flow of ice is defined as a variational inequality, the constraint imposed on the free surface by the bedrock topography is incorporated directly, thus sparing the need for ad hoc post-processing of the free boundary to enforce non-negativity of … 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 Pogorelov [22]). 162, … Theorem 3. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. Marginally trapped surfaces are of central importance in general relativity, where they play the role of apparent horizons, or quasilocal black hole bound-aries. Received 15 September 1979; First Online 01 February 2012; DOI https://doi.org/10.1007/978-3-642-25588-5_15 3 n An. Rational Mech. Department of Mathematics Technical Report19, Lawrence, Kansas: University of Kansas 1968, Chern, S.S., Osserman, R.: Complete minimal surfaces in euclideann-space. Anal.58, 285–307 (1975), Peetre, J.: A generalization of Courant's nodal domain theorem. Ann. Immediate online access to all issues from 2019. Part of Springer Nature. Preprint, Chern, S.S.: Minimal submanifolds in a Riemannian manifold. Many papers have been devoted to investigating stability. And we will study when the sharp isoperimetric inequality for the minimal surface follows from that of the two flat surfaces. 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 Comm. Acad. 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. Rational Mech. %�� The slope inequality asserts that ω2 f ≥ 4g −4 g deg(f∗ωf) for a relative minimal ﬁbration of genus g ≥ 2. n+1 to be isometrically and minimally immersed inM Math. 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˙ Scand.5, 15–20 (1957), Polya, G., Szegö, G.: Isoperimetric inequalities of Mathematical Physics. https://doi.org/10.1007/BF01215521, Over 10 million scientific documents at your fingertips, Not logged in Then!2 X=k 4˜(O X): Let us brie y introduce the history of this inequality. The Sobolev inequality (see Chapter 3). These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively. We note that the construction of the index in this space (in the sense of Fischer-Colbrie [FC85]) in Section 4 is somewhat subtle. Destination page number Search scope Search Text Search scope Search Text Remarks. We note that a noncompact minimal surface is said to be stable if its index is zero. 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. The case involving both charge and angular momentum has been proved recently in [25]. Arch. Minimal surfaces of small total curvature : Martina Jorgensen 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. Pogorelov [22]). << 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. 3. can get a stability-free proof of the slope inequality. Arch. The conjectured Penrose inequality, proved in the Riemannian case by Anal.45, 194–221 (1972), Lawson, Jr., B.: Lectures on Minimal Submanifolds. Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. 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]. It is the curvature characteristic of minimal surfaces that is important. Annals of Mathematics Studies21, Princeton: Princeton, University Press 1951, Simons, J.: Minimal Varieties in Riemannian manifolds. Pure Appl. 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. 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. Deﬁnition 2. By plugging a … J. Math.98, 515–528 (1976), Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. uis minimal. 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. Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Math.10, 271–290 (1957), Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. 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. Let X be a smooth minimal surface of general type over kand of maximal Albanese dimension. Of course the minimal surface will not be stationary for arbitrary changes in the metric. Theorem 1.5 (Severi inequality). Pages 441-456. }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� Stable approximations of a minimal surface problem with variational inequalities Nashed, M. Zuhair; Scherzer, Otmar; Abstract. Classify minimal surfaces in R3 whose Gauss map is … Nonlinear Sampled-Data Systems and Multidimensional Z-Transform (M112) A. Rault and E.I. 3 The Stability Estimate In this section we prove an estimate on the integral of the curvature which will be used in the proof of Bernstein’s theorem. outermost minimal surface is a minimal surface which is not contained entirely inside another minimal surface. Recall that if X is a minimal surface of general type over k, and ω X is the canonical bundle of X, then the Noether inequality asserts that h 0 (ω X) ⩽ 1 2 … Moreover, the minimal model is smooth. Ci. 98, 515–528 (1976) Google Scholar. [SSY], [CS] and [SS]. 68 0 obj interval (δ, 1], where δ > 0, and the extrinsic curvature of the surface satisfies the inequality \K e \ > 3(1 - δ)2/2δ. 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. This is a preview of subscription content, access via your institution. Math. On the other hand, we can use either the Gauss–Bonnet theorem or the Jacobi equation to get the opposite bound. /Filter /FlateDecode Gauss curvature for stable minimal surfaces in R3, which yielded the Bern-stein theorem for complete stable minimal surfaces in R3. 43o ��lʮ��OU�-@6�]U�hj[��2�M�uW�Ũ� ^�t��n�Au��|��x�#*P�,i��˘��. In particular, F(E) F(K) = njKj whenever jEj= jKj. inequality to higher codimension, to non local perimeters and to non euclidean settings such as the Gauss space. Theorem 3.1 ([27, Theorem 0.2]). Circ. 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). Z.144, 169–174 (1975), Departamento de Matematica, Universidade Federal do Ceará, Fortaleza Ceará, Brasil, Instituto de Matematica Pura e Aplicada, Rua Luiz de Camões 68, 20060, Rio de Janeiro, R.J., Brasil, You can also search for this author in $\endgroup$ – User4966 Nov 21 '14 at 7:12 minimal surfaces: Corollary 2. Guisti [3] found nonlinear entire minimal graphs in Rn+1. Math. Math. Stable minimal surfaces have many important properties. 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. Helv.46, 182–213 (1971), Bol, G.: Isoperimetrische Ungleichungen für Bereiche und Flächen. § are C1, parameterized by arclength, such that the tangent vector t = c0 is absolutely continuous. 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. Interestingly, it follows from a stability argument [43] that outermost minimal … Barbosa J.L., Carmo M.. (2012) Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Index, vision number and stability of complete minimal surfaces. Using the inequality of the Lemma for m = 2, we can improve the stability theorem of Barbosa and do Carmo [2]. 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. Z. It is obvious that a complete stable minimal hypersurface in $\mathbb{H}^{n+1}(-1)$ has index 0. Speaker: Chao Xia (Xiamen University) Title: Stability on … Deutsch. the second variation of the area functional is non-negative. Rend. 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). For the … Subscription will auto renew annually. 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. Jber. 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]. Math. 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. 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. These are minimal surfaces which, loosely speaking, are area-minimizing. The key underlying property of the local versions of the inequality is the notion of stability, both for minimal hypersurfaces and for … The UConn Summer School in Minimal Surfaces, Flows, and Relativity is a focused one-week program for graduate students and recent PhDs in geometric analysis, from 16th to 20th, July 2018. Exercise 6. The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. The Zero-Moment Point (ZMP) [1] criterion, namely that 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. Comment. PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of minimal surfaces and eigenvalues of the laplacian. [17, 15]. In the context of multi-contact planning, it was advocated as a generalization ... do not mention how to compute the inequality constraints applying to these new variables. Mech.14, 1049–1056 (1965), Spruck, J.: Remarks on the stability of minimal submanifolds ofR 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. (joint with R. Schoen) Mar 28, 2019 (Thur) 11:00-12:00 @ AB1 502a (Note special date and time.) It is well-known that a minimal graph of codimension one is stable, i.e. To learn the Moser iteration technique, follow [GT]. Learn more about Institutional subscriptions, Barbosa, J.L., do Carmo, M.: On the size of a stable minimal surface inR minimal surface M is a plane (Corollary 4). On the Size of a Stable Minimal Surface in R 3 Pages 115-128. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. $\begingroup$ The problem asks for the stability of the minimal surface. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. Then, the stability inequality reads as R D jr˘j2 +2K˘2 >0. /Length 3024 Math.-Verein.51, 219–257 (1941), Chen, C.C. xڵ�r۸�=_�>U��:��N�u'��&޶3�}�%��$zI�^�}��)��l'm��@>��^�'��x��{�gB�\k9��L0��r��]�f?�8��7N�s��? Barbosa, J. L. (et al.) In fact, all the known proofs are related to some sort of stability: geometric invariant theory in [CH] (in In particular, we consider the space of so-called stable minimal surfaces. the link-to-surface distance) while a ﬁxed contact constraints all six DOFs of the end-effector link. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ(K S ) ( [16]). The isoperimetric inequality for minimal surfaces (see, e.g., Chakerian, Proceedings of the AMS, volume 69, 1978). Ann. so the stability inequality (4) can be written in the form (5) 0 ≤ 2 Z K f2 +4 Z f2 + Z |∇f|2, for any compactly supported function f on F. As noted in Section 2, the ﬁrst eigenvalue of a complete minimal surface in hyperbolic space is bounded below by 1 4. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. Then A 4πQ2 , (43) where A is the area of S and Q is its charge. volume 173, pages13–28(1980)Cite this article. • When S is a K3 surface, Bayer … If (M;g) has positive Ricci curvature, then cannot be stable. In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold M n in the hyperbolic space H n + m in order to show that M n is totally geodesic. minimal surface. Sakrison. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. A minimal surface is called stable if (and only if) the second variation of the area functional is nonnegative for all compactly supported deformations. of Math.40, 834–854 (1939), Smale, S.: On the Morse index theorem. Amer. 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. ... 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. 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. Stable approximations of a minimal surface problem with variational inequalities. 1 In [16] the expected inequality for area and charge has been proved for stable minimal surfaces on time symmetric initial data. References : Complete minimal surfaces with total curvature −2π. For the minimal surface problem associated with (6) – (7), it is shown in Section 4 of Chapter … Jaigyoung Choe's main interest is in differential geometry. Math Z 173, 13–28 (1980). ... J. Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, MSRI preprint. 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. If is a stable minimal … 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 a time symmetric Cauchy surface, then θ+ = 0 if and only if Σ is minimal. Mini-courses will be given by. Again, there is a chosen end of M3, and “contained entirely inside” is deﬁned with respect to this end. 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. Otis Chodosh (Princeton University) Geometric features of the Allen-Cahn equation; Ailana Fraser (University of British Columbia) Minimal surface methods in geometry Arch. Let S be a stable minimal surface. Proof. Jury. A Stability Inequality For a Class of Nonlinear Feedback Systems (M114) A.G. Dewey. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is ﬂat. (to appear), Bandle, C.: Konstruktion isomperimetrischer Ungleichungen der Mathematischen Physik aus solchen der Geometrie.
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