Get quick answers to your questions about the article from our AI researcher chatbot
{'id': 'https://openalex.org/W2964088204', 'doi': 'https://doi.org/10.1090/tran/7281', 'title': 'The extension and convergence of mean curvature flow in higher codimension', 'display_name': 'The extension and convergence of mean curvature flow in higher codimension', 'publication_year': 2017, 'publication_date': '2017-10-04', 'ids': {'openalex': 'https://openalex.org/W2964088204', 'doi': 'https://doi.org/10.1090/tran/7281', 'mag': '2964088204'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/tran/7281', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'acceptedVersion', 'is_accepted': True, 'is_published': False}, 'type': 'article', 'type_crossref': 'journal-article', 'indexed_in': ['crossref'], 'open_access': {'is_oa': True, 'oa_status': 'bronze', 'oa_url': 'https://doi.org/10.1090/tran/7281', 'any_repository_has_fulltext': True}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5113540335', 'display_name': 'Kefeng Liu', 'orcid': None}, 'institutions': [{'id': 'https://openalex.org/I161318765', 'display_name': 'University of California, Los Angeles', 'ror': 'https://ror.org/046rm7j60', 'country_code': 'US', 'type': 'education', 'lineage': ['https://openalex.org/I161318765']}, {'id': 'https://openalex.org/I76130692', 'display_name': 'Zhejiang University', 'ror': 'https://ror.org/00a2xv884', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76130692']}], 'countries': ['CN', 'US'], 'is_corresponding': False, 'raw_author_name': 'Kefeng Liu', 'raw_affiliation_strings': ["Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China – and – Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California, 90095-1555"], 'affiliations': [{'raw_affiliation_string': "Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China – and – Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California, 90095-1555", 'institution_ids': ['https://openalex.org/I161318765', 'https://openalex.org/I76130692']}]}, {'author_position': 'middle', 'author': {'id': 'https://openalex.org/A5101076778', 'display_name': 'Hongwei Xu', 'orcid': 'https://orcid.org/0009-0006-0809-6918'}, 'institutions': [{'id': 'https://openalex.org/I76130692', 'display_name': 'Zhejiang University', 'ror': 'https://ror.org/00a2xv884', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76130692']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Hongwei Xu', 'raw_affiliation_strings': ["Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China"], 'affiliations': [{'raw_affiliation_string': "Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China", 'institution_ids': ['https://openalex.org/I76130692']}]}, {'author_position': 'middle', 'author': {'id': 'https://openalex.org/A5100357221', 'display_name': 'Fei Ye', 'orcid': 'https://orcid.org/0000-0002-5894-2178'}, 'institutions': [{'id': 'https://openalex.org/I76130692', 'display_name': 'Zhejiang University', 'ror': 'https://ror.org/00a2xv884', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76130692']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Fei Ye', 'raw_affiliation_strings': ["Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China"], 'affiliations': [{'raw_affiliation_string': "Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China", 'institution_ids': ['https://openalex.org/I76130692']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5067392577', 'display_name': 'Entao Zhao', 'orcid': 'https://orcid.org/0000-0002-6954-6664'}, 'institutions': [{'id': 'https://openalex.org/I76130692', 'display_name': 'Zhejiang University', 'ror': 'https://ror.org/00a2xv884', 'country_code': 'CN', 'type': 'education', 'lineage': ['https://openalex.org/I76130692']}], 'countries': ['CN'], 'is_corresponding': False, 'raw_author_name': 'Entao Zhao', 'raw_affiliation_strings': ["Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China"], 'affiliations': [{'raw_affiliation_string': "Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People's Republic of China", 'institution_ids': ['https://openalex.org/I76130692']}]}], 'institution_assertions': [], 'countries_distinct_count': 2, 'institutions_distinct_count': 2, 'corresponding_author_ids': [], 'corresponding_institution_ids': [], 'apc_list': None, 'apc_paid': None, 'fwci': 2.449, 'has_fulltext': True, 'fulltext_origin': 'ngrams', 'cited_by_count': 20, 'citation_normalized_percentile': {'value': 0.695698, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 90, 'max': 91}, 'biblio': {'volume': '370', 'issue': '3', 'first_page': '2231', 'last_page': '2262'}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T10229', 'display_name': 'Geometric Analysis and Curvature Flows', 'score': 0.9999, 'subfield': {'id': 'https://openalex.org/subfields/2604', 'display_name': 'Applied Mathematics'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, 'topics': [{'id': 'https://openalex.org/T10229', 'display_name': 'Geometric Analysis and Curvature Flows', 'score': 0.9999, 'subfield': {'id': 'https://openalex.org/subfields/2604', 'display_name': 'Applied Mathematics'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T12351', 'display_name': 'Geometry and complex manifolds', 'score': 0.9996, 'subfield': {'id': 'https://openalex.org/subfields/2608', 'display_name': 'Geometry and Topology'}, 'field': {'id': 'https://openalex.org/fields/26', 'display_name': 'Mathematics'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}, {'id': 'https://openalex.org/T13080', 'display_name': 'Advanced Differential Geometry Research', 'score': 0.992, 'subfield': {'id': 'https://openalex.org/subfields/3103', 'display_name': 'Astronomy and Astrophysics'}, 'field': {'id': 'https://openalex.org/fields/31', 'display_name': 'Physics and Astronomy'}, 'domain': {'id': 'https://openalex.org/domains/3', 'display_name': 'Physical Sciences'}}], 'keywords': [], 'concepts': [{'id': 'https://openalex.org/C127813850', 'wikidata': 'https://www.wikidata.org/wiki/Q6803617', 'display_name': 'Mean curvature flow', 'level': 4, 'score': 0.6444402}, {'id': 'https://openalex.org/C195065555', 'wikidata': 'https://www.wikidata.org/wiki/Q214881', 'display_name': 'Curvature', 'level': 2, 'score': 0.56772786}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.5160937}, {'id': 'https://openalex.org/C2777303404', 'wikidata': 'https://www.wikidata.org/wiki/Q759757', 'display_name': 'Convergence (economics)', 'level': 2, 'score': 0.5057492}, {'id': 'https://openalex.org/C151300846', 'wikidata': 'https://www.wikidata.org/wiki/Q125539', 'display_name': 'Submanifold', 'level': 2, 'score': 0.50018}, {'id': 'https://openalex.org/C11413529', 'wikidata': 'https://www.wikidata.org/wiki/Q8366', 'display_name': 'Algorithm', 'level': 1, 'score': 0.49571407}, {'id': 'https://openalex.org/C175017881', 'wikidata': 'https://www.wikidata.org/wiki/Q1318998', 'display_name': 'Mean curvature', 'level': 3, 'score': 0.33165836}, {'id': 'https://openalex.org/C2524010', 'wikidata': 'https://www.wikidata.org/wiki/Q8087', 'display_name': 'Geometry', 'level': 1, 'score': 0.30938977}, {'id': 'https://openalex.org/C134306372', 'wikidata': 'https://www.wikidata.org/wiki/Q7754', 'display_name': 'Mathematical analysis', 'level': 1, 'score': 0.23615152}, {'id': 'https://openalex.org/C162324750', 'wikidata': 'https://www.wikidata.org/wiki/Q8134', 'display_name': 'Economics', 'level': 0, 'score': 0.0}, {'id': 'https://openalex.org/C50522688', 'wikidata': 'https://www.wikidata.org/wiki/Q189833', 'display_name': 'Economic growth', 'level': 1, 'score': 0.0}], 'mesh': [], 'locations_count': 2, 'locations': [{'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/tran/7281', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'acceptedVersion', 'is_accepted': True, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'http://arxiv.org/abs/1104.0971', 'pdf_url': 'http://arxiv.org/pdf/1104.0971', 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I205783295', 'host_organization_name': 'Cornell University', 'host_organization_lineage': ['https://openalex.org/I205783295'], 'host_organization_lineage_names': ['Cornell University'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://doi.org/10.1090/tran/7281', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S60030702', 'display_name': 'Transactions of the American Mathematical Society', 'issn_l': '0002-9947', 'issn': ['0002-9947', '1088-6850'], 'is_oa': False, 'is_in_doaj': False, 'is_core': True, 'host_organization': 'https://openalex.org/P4310315719', 'host_organization_name': 'American Mathematical Society', 'host_organization_lineage': ['https://openalex.org/P4310315719'], 'host_organization_lineage_names': ['American Mathematical Society'], 'type': 'journal'}, 'license': 'publisher-specific-oa', 'license_id': 'https://openalex.org/licenses/publisher-specific-oa', 'version': 'acceptedVersion', 'is_accepted': True, 'is_published': False}, 'sustainable_development_goals': [], 'grants': [{'funder': 'https://openalex.org/F4320306076', 'funder_display_name': 'National Science Foundation', 'award_id': '11201416'}, {'funder': 'https://openalex.org/F4320306076', 'funder_display_name': 'National Science Foundation', 'award_id': '11531012'}, {'funder': 'https://openalex.org/F4320306076', 'funder_display_name': 'National Science Foundation', 'award_id': '11371315'}], 'datasets': [], 'versions': [], 'referenced_works_count': 33, 'referenced_works': ['https://openalex.org/W1491650242', 'https://openalex.org/W1509346826', 'https://openalex.org/W1569099763', 'https://openalex.org/W1605703710', 'https://openalex.org/W1964899169', 'https://openalex.org/W1977333154', 'https://openalex.org/W1982425238', 'https://openalex.org/W2006533958', 'https://openalex.org/W2021350139', 'https://openalex.org/W2023139263', 'https://openalex.org/W2034752215', 'https://openalex.org/W2045738464', 'https://openalex.org/W2069026431', 'https://openalex.org/W2077299535', 'https://openalex.org/W2082052570', 'https://openalex.org/W2082618032', 'https://openalex.org/W2086609809', 'https://openalex.org/W2089311377', 'https://openalex.org/W2093041805', 'https://openalex.org/W2100471299', 'https://openalex.org/W2102314081', 'https://openalex.org/W2104360622', 'https://openalex.org/W2122172735', 'https://openalex.org/W2131183115', 'https://openalex.org/W2137627495', 'https://openalex.org/W2161010955', 'https://openalex.org/W2963481257', 'https://openalex.org/W2963555594', 'https://openalex.org/W3022122637', 'https://openalex.org/W3106310623', 'https://openalex.org/W3151501167', 'https://openalex.org/W4212837841', 'https://openalex.org/W623266445'], 'related_works': ['https://openalex.org/W4304163146', 'https://openalex.org/W4297825487', 'https://openalex.org/W2964267756', 'https://openalex.org/W2788820315', 'https://openalex.org/W2735504001', 'https://openalex.org/W2508030851', 'https://openalex.org/W2161845662', 'https://openalex.org/W2160481556', 'https://openalex.org/W1597471746', 'https://openalex.org/W1481929931'], 'abstract_inverted_index': {'In': [0], 'this': [1], 'paper,': [2], 'we': [3], 'investigate': [4], 'the': [5, 8, 51, 62], 'convergence': [6], 'of': [7, 12], 'mean': [9, 63], 'curvature': [10, 58, 64], 'flow': [11, 65], 'closed': [13], 'submanifolds': [14], 'in': [15, 73], '<inline-formula': [16], 'content-type="math/mathml">': [17], '<mml:math': [18], 'xmlns:mml="http://www.w3.org/1998/Math/MathML"': [19], 'alttext="double-struck': [20], 'upper': [21], 'R': [22], 'Superscript': [23], 'n': [24], 'plus': [25], 'q">': [26], '<mml:semantics>': [27], '<mml:msup>': [28], '<mml:mrow': [29, 34], 'class="MJX-TeXAtom-ORD">': [30, 35], '<mml:mi': [31], 'mathvariant="double-struck">R</mml:mi>': [32], '</mml:mrow>': [33, 39], '<mml:mi>n</mml:mi>': [36], '<mml:mo>+</mml:mo>': [37], '<mml:mi>q</mml:mi>': [38], '</mml:msup>': [40], '<mml:annotation': [41], 'encoding="application/x-tex">\\mathbb': [42], '{R}^{n+q}</mml:annotation>': [43], '</mml:semantics>': [44], '</mml:math>': [45], '</inline-formula>.': [46], 'We': [47], 'show': [48], 'that': [49], 'if': [50], 'initial': [52], 'submanifold': [53], 'satisfies': [54], 'some': [55], 'suitable': [56], 'integral': [57], 'conditions,': [59], 'then': [60], 'along': [61], 'it': [66], 'will': [67], 'shrink': [68], 'to': [69], 'a': [70], 'round': [71], 'point': [72], 'finite': [74], 'time.': [75]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W2964088204', 'counts_by_year': [{'year': 2024, 'cited_by_count': 1}, {'year': 2023, 'cited_by_count': 2}, {'year': 2021, 'cited_by_count': 2}, {'year': 2020, 'cited_by_count': 7}, {'year': 2019, 'cited_by_count': 1}, {'year': 2015, 'cited_by_count': 2}, {'year': 2014, 'cited_by_count': 1}, {'year': 2013, 'cited_by_count': 2}, {'year': 2012, 'cited_by_count': 1}], 'updated_date': '2025-01-08T02:51:46.288641', 'created_date': '2019-07-30'}